Neoproterozoic geodynamic evolution of SW-Gondwana: a southern African perspective

Our current understanding of the tectonic history of the principal Pan-African orogenic belts in southwestern Africa, reaching from the West Congo Belt in the north to the Lufilian/Zambezi, Kaoko, Damara, Gariep and finally the Saldania Belt in the south, is briefly summarized. On that basis, possible links with tectono-stratigraphic units and major structures on the eastern side of the Rio de la Plata Craton are suggested, and a revised geodynamic model for the amalgamation of SW-Gondwana is proposed. The Rio de la Plata and Kalahari Cratons are considered to have become juxtaposed already by the end of the Mesoproterozoic. Early Neoproterozoic rifting led to the fragmentation of the northwestern (in today`s coordinates) Kalahari Craton and the splitting off of several small cratonic blocks. The largest of these ex-Kalahari cratonic fragments is probably the Angola Block. Smaller fragments include the Luis Alves and Curitiba microplates in eastern Brazil, several basement inliers within the Damara Belt, and an elongate fragment off the western margin, named Arachania. The main suture between the Kalahari and the Congo-So Francisco Cratons is suspected to be hidden beneath younger cover between the West Congo Belt and the Lufilian/Zambezi Belts and probably continues westwards via the Cabo Frio Terrane into the Goias magmatic arc along the Brasilia Belt. Many of the rift grabens that separated the various former Kalahari cratonic fragments did not evolve into oceanic basins, such as the Northern Nosib Rift in the Damara Belt and the Gariep rift basin. Following latest Cryogenian/early Ediacaran closure of the Brazilides Ocean between the Rio de la Plata Craton and the westernmost fragment of the Kalahari Craton, the latter, Arachania, became the locus of a more than 1,000-km-long continental magmatic arc, the Cuchilla Dionisio-Pelotas Arc. A correspondingly long back-arc basin (Marmora Basin) on the eastern flank of that arc is recognized, remnants of which are found in the Marmora Terrane-the largest accumulation of oceanic crustal material known from any of the Pan-African orogenic belts in the region. Corresponding foredeep deposits that emerged from the late Ediacaran closure of this back-arc basin are well preserved in the southern areas, i.e. the Punta del Este Terrane, the Marmora Terrane and the Tygerberg Terrane. Further to the north, present erosion levels correspond with much deeper crustal sections and comparable deposits are not preserved anymore. Closure of the Brazilides Ocean, and in consequence of the Marmora back-arc basin, resulted from a change in the Rio de la Plata plate motion when the Iapetus Ocean opened between the latter and Laurentia towards the end of the Ediacaran. Later break-up of Gondwana and opening of the modern South Atlantic would have followed largely along the axis of the Marmora back-arc basin and not along major continental sutures.

High T/P evolution and metamorphic ages of the migmatitic basement of northern Sierras Pampeanas, Argentina: Characterization of a mid-crustal segment of the Famatinian belt

New petrologic, thermobarometric and U-Pb monazite geochronologic information allowed to resolve the metamorphic evolution of a high temperature mid-crustal segment of an ancient subduction-related orogen. The EI Portezuelo Metamorphic-Igneous Complex, in the northern Sierras Pampeanas, is mainly composed of migmatites that evolved from amphibolite to granulite metamorphic facies, reaching thermal peak conditions of 670-820 degrees C and 4.5-5.3 kbar. The petrographic study combined with conventional and pseudosection thermobarometry led to deducing a short prograde metamorphic evolution within migmatite blocks. The garnet-absent migmatites represent amphibolite-facies rocks, whereas the cordierite-garnet-K-feldspar-sillimanite migmatites represent higher metamorphic grade rocks. U-Pb geochronology on monazite grains within leucosome record the time of migmatization between approximate to 477 and 470 Ma. Thus, the El Portezuelo Metamorphic-Igneous Complex is an example of exhumed Early Ordovician anatectic middle crust of the Famatinian mobile belt. Homogeneous exposure of similar paleo-depths throughout the Famatinian back-arc and isobaric cooling paths suggest slow exhumation and consequent longstanding crustal residence at high temperatures. High thermal gradients uniformly distributed in the Famatinian back-arc can be explained by shallow convection of a low-viscosity asthenosphere promoted by subducting-slab dehydration. (C) 2011 Elsevier Ltd. All rights reserved.

The Cariris Velhos tectonic event in Northeast Brazil

The Borborema Province in northeastern South America is a typical Brasiliano-Pan-African branching system of Neoproterozoic orogens that forms part of the Western Gondwana assembly. The province is positioned between the Sao Luis-West Africa craton to the north and the Sao Francisco (Congo-Kasai) craton to the south. For this province the main characteristics are (a) its subdivision into five major tectonic domains, bounded mostly by long shear zones, as follows: Medio Coreau, Ceara Central, Rio Grande do Norte, Transversal, and Southern; (b) the alternation of supracrustal belts with reworked basement inliers (Archean nuclei + Paleoproterozoic belts); and (c) the diversity of granitic plutonism, from Neoproterozoic to Early Cambrian ages, that affect supracrustal rocks as well as basement inliers. Recently, orogenic rock assemblages of early Tonian (1000-920 Ma) orogenic evolution have been recognized, which are restricted to the Transversal and Southern domains of the Province. Within the Transversal Zone, the Alto Pajeu terrane locally includes some remnants of oceanic crust along with island arc and continental arc rock assemblages, but the dominant supracrustal rocks are mature and immature pelitic metasedimentary and metavolcaniclastic rocks. Contiguous and parallel to the Alto Pajeu terrane, the Riacho Gravata subterrane consists mainly of low-grade metamorphic successions of metarhythmites, some of which are clearly turbiditic in origin, metaconglomerates, and sporadic marbles, along with interbedded metarhyolitic and metadacitic volcanic or metavolcaniclastic rocks. Both terrane and subterrane are cut by syn-contractional intrusive sheets of dominantly peraluminous high-K calc-alkaline, granititic to granodioritic metaplutonic rocks. The geochemical patterns of both supracrustal and intrusive rocks show similarities with associations of mature continental arc volcano-sedimentary sequences, but some subordinate intra-plate characteristics are also found. In both the Alto Pajeu and Riacho Gravata terranes, TIMS and SHRIMP U-Pb isotopic data from zircons from both metavolcanic and metaplutonic rocks yield ages between 1.0 and 0.92 Ga, which define the time span for an event of orogenic character, the Cariris Velhos event. Less extensive occurrences of rocks of Cariris Velhos age are recognized mainly in the southernmost domains of the Province, as for example in the Polo Redondo-Maranco terrane, where arc-affinity migmatite-granitic and meta-volcano-sedimentary rocks show U-Pb ages (SHRIMP data) around 0.98-0.97 Ga. For all these domains, Sm-Nd data exhibit Tom model ages between 1.9 and 1.1 Ga with corresponding slightly negative to slightly positive epsilon(Nd)(t) values. These domains, along with the Borborema Province as a whole, were significantly affected by tectonic and magmatic events of the Brasiliano Cycle (0.7-0.5 Ga), so that it is possible that there are some other early Tonian rock assemblages which were completely masked and hidden by these later Brasiliano events. Cariris Velhos processes are younger than the majority of orogenic systems at the end of Mesoproterozoic Era and beginning of Neoproterozoic throughout the world, e.g. Irumide belt, Kibaride belt and Namaqua-Natal belt, and considerably younger than those of the youngest orogenic process (Ottawan) in the Grenvillian System. Therefore, they were probably not associated with the proposed assembly of Rodinia. We suggest, instead, that Cariris Velhos magmatism and tectonism could have been related to a continental margin magmatic arc, with possible back-arc associations, and that this margin may have been a short-lived (<100 m.y.) leading edge of the newly assembled Rodinia supercontinent. (C) 2009 Elsevier Ltd. All rights reserved.

Boron isotope composition of tourmalinite and vein tourmalines associated with gold mineralization, Serra do Itaberaba Group, central Ribeira Belt, SE Brazil

Important concentrations of tourmaline occur as gold-bearing stratiform tourmalinites and in mineralized quartz-tourmaline veins at the Tapera Grande and Quartzito gold prospects in the Mesoproterozoic Serra do Itaberaba Group, central Ribeira Belt (Sao Paulo State, SE Brazil). The main rock types in both prospects constitute the volcanic-sedimentary Morro da Pedra Preta Formation, which formed in a submarine back-arc setting. At Tapera Grande, the volcanic-sedimentary sequence is composed of metabasic and metavolcaniclastic rocks, graphitic and sulfur-rich metapelites, banded iron formation, metandesite, metarhyolite, calcsilicates, tourmalinites and metahydrothermalites derived from mafic and felsic rocks. The Mesoproterozoic rocks at Quartzito prospect are lithologically similar but they have been affected by Neoproterozoic faulting and shearing and by the emplacement of granitic rocks, resulting in the formation of tourmaline-rich quartz-carbonate veins with gold and base metal mineralization. We conducted a chemical and B-isotope study of tourmalines in order to better understand the origin of the stratiform tourmalinites in the Morro da Pedra Preta Formation and their relationship with gold mineralization. The overall range of delta(11)B values obtained for the tourmalinite and vein tourmalines is between - 15%. and -5 parts per thousand, with the tourmalinites failing at the low end of this range (-15 to -8 parts per thousand). Such values are typical for continental crust and inconsistent with a primary marine boron signature as expected from the submarine-exhalative model for the gold prospects. We conclude from this that tourmaline formed or recrystallized from crustal fluids related to the amphibolite-grade metamorphism which affected the Serra do Itaberaba Group and that gold deposition occurred syn- to post-peak metamorphism by phase immiscibility, as attested by fluid inclusions in Tapera Grande tourmalinite tourmaline and quartz. The vein-hosted tourmalines at Quartzito have isotopically variable boron signatures, with heavier delta(11)B values of -5 parts per thousand to -8 parts per thousand for acicular green tourmalines and lighter values (-15 parts per thousand to -7 parts per thousand for light blue, Ti-firee tourmaline from quartz-carbonate veins). We attribute the heavier boron to fluids derived from the volcano-sedimentary rocks of marine affinity whereas the lighter boron was contributed by crustal fluids related to the granitoids or metasediments in the continental crust. (c) 2009 Elsevier B.V. All rights reserved.

U-Pb-Hf-trace element systematics and geochronology of zircon from a granulite-facies metamorphosed mafic-ultramafic layered complex in Central Brazil

Zircon recrystallization is a common process during high-grade metamorphism and promotes partial or complete resetting of the original isotopic and chemical characteristics of the mineral and thus complicates U-Pb geochronological interpretation. In Central Brazil, this complexity may be illustrated by three composite mafic-ultramafic intrusions metamorphosed under amphibolite-to-granulite conditions. Their ages of emplacement and metamorphic ages have been a matter of controversy for the last thirty years. The Serra da Malacacheta and Barro Alto complexes make up the southernmost of these layered bodies and four samples from distinct rock types were investigated in order to verify the consequences of metamorphic alteration of zircon for U-Pb dating. Cathodoluminescent imaging reveals internal features which are typical of concomitant dissolution-reprecipitation processes, such as convolute zoning and inward-moving recrystallization fronts, even in samples in which partially preserved igneous textures are observed. Due to this extensive alteration, LA-ICPMS U-Pb isotopic analysis yielded inconclusive data. However, in situ Hf isotopic and trace-element analyses help to clarify the real meaning of the geochronological data. Low Lu/Hf (<0.004) and homogeneous (176)Hf/(177)Hf(t) values imply that the zircon populations within individual samples have crystallized in a single episode, despite the observed variations in age values. Trace element signatures of zircon grains from garnet-bearing samples reveal that they were unreactive to the development of the peak metamorphism mineral assemblage and, thus, the main chemical feature in such grains is attributed to a coupled dissolution-reprecipitation process. However, in the Cafelandia amphibolite an additional alteration process is identified, probably related to the influx of late-stage fluids. Combined isotopic and geochemical investigation on zircon grains allowed the distinction of two magmatic events. The first corresponds to the crystallization of the Serra da Malacacheta Complex and characterizes a juvenile magmatism at similar to 1.3 Ga. The younger episode, recognized in the Barro Alto Complex, is dated at ca. 800 Ma and is represented by mafic and ultramafic rocks showing intense contamination with continental crust, implying that the emplacement took place, most likely, in a continental back-arc setting. Altered zircon domains as well as titanite grains date the metamorphic event at ca. 760-750 Ma. (C) 2011 Elsevier B.V. All rights reserved.

Tectonic implications of a paleomagnetic study of the Sarmiento Ophiolitic Complex, southern Chile

A paleomagnetic study was carried out on the Late Jurassic Sarmiento Ophiolitic Complex (SOC) exposed in the Magallanes fold and thrust belt in the southern Patagonian Andes (southern Chile). This complex, mainly consisting of a thick succession of pillow-lavas, sheeted dikes and gabbros, is a seafloor remnant of the Late Jurassic to Early Cretaceous Rocas Verdes basin that developed along the south-western margin of South America. Stepwise thermal and alternating field demagnetization permitted the isolation of a post-folding characteristic remanence, apparently carried by fine grain (SD?) magnetite, both in the pillow-lavas and dikes. The mean ""in situ"" direction for the SOC is Dec: 286.9 degrees, Inc: -58.5 degrees, alpha-95: 6.9 degrees, N: 11 (sites). Rock magnetic properties, petrography and whole-rock K-Ar ages in the same rocks are interpreted as evidence of correlation between remanence acquisition and a greenschist facies metamorphic overprint that must have occurred during latest stages or after closure and tectonic inversion of the basin in the Late Cretaceous. The mean remanence direction is anomalous relative to the expected Late Cretaceous direction from stable South America. Particularly, a declination anomaly over 50 degrees is suggestively similar to paleomagnetically interpreted counter clockwise rotations found in thrust slices of the Jurassic El Quemado Fm. located over 100 km north of the study area in Argentina. Nevertheless, a significant ccw rotation of the whole SOC is difficult to reconcile with geologic evidence and paleogeographic models that suggest a narrow back-arc basin sub-parallel to the continental margin. A rigid-body 30 degrees westward tilting of the SOC block around a horizontal axis trending NNW, is considered a much simpler explanation, being consistent with geologic evidence. This may have occurred as a consequence of inverse reactivation of old normal faults, which limit both the SOC exposures and the Cordillera Sarmiento to the East. The age of tilting is unknown but it must postdate remanence acquisition in the Late Cretaceous. Two major orogenic events of the southern Patagonian Andes, in the Eocene (ca. 42 Ma) and Middle Miocene (ca. 12 Ma), respectively, could have caused the proposed tilting. (C) 2008 Elsevier B.V. All rights reserved.

Precambrian geotectonic units of the Rio de La Plata craton

The main Precambrian tectonic units of Uruguay include the Piedra Alta tectonostratigraphic terrane (PATT) and Nico Perez tectonostratigraphic terrane (NPTT), separated by the Sarandi del Yi high-strain zone. Both terranes are well exposed in the Rio de La Plata craton (RPC). Although these tectonic units are geographically small, they record a wide span of geologic time. Therefore improved geological knowledge of this area provides a fuller understanding of the evolution of the core of South America. The PATT is constituted by low-to medium-grade metamorphic belts (ca. 2.1 Ga); its petrotectonic associations such as metavolcanic units, conglomerates, banded iron formations, and turbiditic deposits suggest a back-arc or a trench-basin setting. Also in the PATT, a late to post-orogenic, arc-related layered mafic complex (2.3-1.9 Ga), followed by A-type granites (2.08 Ga), and finally a taphrogenic mafic dike swarm (1.78 Ga) occur. The less thoroughly studied NPTT consists of Palaeoproterozoic high-grade metamorphic sequences (ca. 2.2 Ga), mylonites and postorogenic and rapakivi granites (1.75 Ga). The Brasiliano-Pan African orogeny affected this terrane. Neoproterozoic cover occurs in both tectonostratigraphic terranes, but is more developed in the NPTT. Over the past 15 years, new isotopic studies have improved our recognition of different tectonic events and associated processes, such as reactivation of shear zones and fluids circulation. Transamazonian and Statherian tectonic events were recognized in the RPC. Based on magmatism, deformation, basin development and metamorphism, we propose a scheme for the Precambrian tectonic evolution of Uruguay, which is summarized in the first Palaeoproterozoic tectonic map of the Rio de La Plata craton.

Petrology and geochemistry of the banded iron formation in the Eastern Sierras Pampeanas of San Luis (Argentina): Implications for the evolution of the Nogoli Metamorphic Complex

The metamorphosed banded iron formation from the Nogoli Metamorphic Complex of western Sierra de San Luis, Eastern Sierras Pampeanas of Argentina (Nogoli area, 32 degrees 55`S-66 degrees 15`W) is classified as an oxide facies iron formation of Algoma Type, with a tectonic setting possibly associated with an island arc or back arc, on the basis of field mapping, mineral and textural arrangements and whole rock geochemical features. The origin of banded iron formation is mainly related to chemical precipitation of hydrogenous sediments from seawater in oceanic environments. The primary chemical precipitate is a result of solutions that represent mixtures of seawater and hydrothermal fluids, with significant dilution by maficultramafic volcanic and siliciclastic materials. Multi-stage T(DM) model ages of 1670, 1854 and 1939 Ma and positive, mantle-like xi Nd((1502)) values of +3.8, +1.5 and +0.5 from the banded iron formation are around the range of those mafic to ultramafic meta-volcanic rocks of Nogoli Metamorphic Complex, which are between 1679 and 1765 Ma and +2.64 and +3.68, respectively. This Sm and Nd isotopic connection suggests a close genetic relationship between ferruginous and mafic-ultramafic meta-volcanic rocks, as part of the same island arc or back arc setting. A previous Sm-Nd whole rock isochron of similar to 1.5 Ga performed on mafic-ultramafic meta-volcanic rocks led to the interpretation that chemical sedimentation as old as Mesoproterozoic is possible for the banded iron formation. A clockwise P-T path can be inferred for the regional metamorphic evolution of the banded iron formation, with three distinctive trajectories: (1) Relict prograde M(1)-M(3) segment with gradual P and T increase from greenschist facies at M(1) to amphibolite facies at M(3). (2) Peak P-T conditions at high amphibolite-low granulite facies during M(4). (3) Retrograde counterpart of M(4), that returns from amphibolite facies and stabilizes at greenschist facies during M(5). Each trajectory may be regarded as produced by different tectonic events related to the Pampean? (1) and the Famatinian (2 and 3) orogenies, during the Early to Middle Paleozoic. The Nogoli Metamorphic Complex is interpreted as part of a greenstone belt within the large Meso- to Neoproterozoic Pampean Terrane of the Eastern Sierras Pampeanas of Argentina. (C) 2009 Elsevier Ltd. All rights reserved.

Tissue inflammatory response to implantation of calcium hydroxide and iodoform in the back of rats

PURPOSE: This study evaluated the inflammatory reaction caused by the implantation of iodoform and calcium hydroxide in the back of rats. These drugs may be used as intracanal dressings to eliminate residual bacteria of the root canal system. METHODS: Twenty albinic rats (Rattus norvegicus, var Wistar) were divided into four groups: control group 1 (CG1) had normal skin; control group 2 (CG2) had wounded tissue without drugs; in groups 3 and 4, iodoform (IG) and calcium hydroxide (CHG) were inserted into the wounds, respectively. After 3, 5 and 11 days, slices of the implanted areas were macroscopically and microscopically observed regarding to their qualitative and quantitative aspects. RESULTS: In the macroscopical analysis, the CHG showed a large area of necrosis and swelling, which progressively decreased; in the IG the presence of iodoform surrounded by normal tissue was observed. The qualitative and quantitative histological analysis showed that IG promoted a shorter delay in the inflammatory response than the CHG. CONCLUSION: The inflammatory reaction for iodoform had a peak period five days after the drug insertion. By comparison, calcium hydroxide showed a very large area of necrosis that could only be partially eliminated after eleven days.

An Extension of the Invariance Principle for a Class of Differential Equations with Finite Delay

An extension of the uniform invariance principle for ordinary differential equations with finite delay is developed. The uniform invariance principle allows the derivative of the auxiliary scalar function V to be positive in some bounded sets of the state space while the classical invariance principle assumes that. V <= 0. As a consequence, the uniform invariance principle can deal with a larger class of problems. The main difficulty to prove an invariance principle for functional differential equations is the fact that flows are defined on an infinite dimensional space and, in such spaces, bounded solutions may not be precompact. This difficulty is overcome by imposing the vector field taking bounded sets into bounded sets.

Evaluation of Cadaveric Renal Vein Lengths and Their Extension Loss with Three Types of Ligature and Section

Background and Purpose: The right kidney has been less frequently used in live donor nephrectomy, because of the shorter length of the right renal vein (RRV) that is associated with technical difficulties and higher rates of venous thrombosis. In live open donor or deceased donor transplant nephrectomy, an additional cuff of the inferior vena cava is usually removed, but this is a more difficult and risky maneuver in laparoscopic nephrectomy. For this reason, laparoscopic right nephrectomy (LRN) for renal transplantation (RT) is not frequently performed in most medical institutions. We evaluate the difference between RRV and left renal vein (LRV) lengths in cadavers, as harvested for RT by three clamping methods. Our objective was to obtain information that could clarify when LRN for RT should be encouraged or avoided with regard to conventional surgery. Materials and Methods: Ninety adult fresh unfrozen cadavers were randomly divided into three groups of 30, according to the clamping device used: Satinsky, stapler, and Hem-o-lok clip. The abdominal viscera were removed through a median xyphopubic incision, and the veins were measured on the bench. Two lateral limits were used: The renal hilum and the tangential line of the renal poles. As for medial limits, the inferior vena cava or the laparoscopic clipping device on the RRV were used on the right side, while on the LRV, the medial border of the emergence of the adrenal vein was considered. After section of the renal vein, a slight traction of the extremity was applied for the measurement. All measurements were obtained three times using a metallic millimetric ruler, and the arithmetic mean was considered. The chi-square, one-way analysis of variance, and paired t tests were used for statistical analysis. Statistical significance was accepted at P <= 0.05. Results: The groups of cadavers were homogeneous in demographic characteristics. Regardless of the clamping method and considering the useful length of the LRV, the RRV was statistically smaller. The evaluation of the vein length did not depend on the lateral limit considered. Independent of the clamping method, on both sides, the lengths after the vein section were larger than before the section, a fact attributed to traction. Use of a stapler and a single Hem-o-lok presented the same waste of vein length on the right side. On average, the RRV was 13.7% shorter than the LRV. Conclusions: With the wide acceptance of laparoscopic live donor nephrectomy, the length difference between the veins of both kidneys is an important issue, and the right kidney is therefore used less than the left, compared with conventional surgery. This article represents the first step to quantify the anatomic length of renal veins in different situations. Certainly, more imagenologic or surgical studies should be carried out before decisions can be made for better selection of patients for LRN.

Surface plasmon resonance of gold nanoparticles formed by cathodic arc plasma ion implantation into polymer

Shallow subsurface layers of gold nanoclusters were formed in polymethylmethacrylate (PMMA) polymer by very low energy (49 eV) gold ion implantation. The ion implantation process was modeled by computer simulation and accurately predicted the layer depth and width. Transmission electron microscopy (TEM) was used to image the buried layer and individual nanoclusters; the layer width was similar to 6-8 nm and the cluster diameter was similar to 5-6 nm. Surface plasmon resonance (SPR) absorption effects were observed by UV-visible spectroscopy. The TEM and SPR results were related to prior measurements of electrical conductivity of Au-doped PMMA, and excellent consistency was found with a model of electrical conductivity in which either at low implantation dose the individual nanoclusters are separated and do not physically touch each other, or at higher implantation dose the nanoclusters touch each other to form a random resistor network (percolation model). (C) 2009 American Vacuum Society. [DOI: 10.1116/1.3231449]

Experimental studies of di-jet survival and surface emission bias in Au plus Au collisions via angular correlations with respect to back-to-back leading hadrons

We report first results from an analysis based on a new multi-hadron correlation technique, exploring jet-medium interactions and di-jet surface emission bias at the BNL Relativistic Heavy Ion Collider (RHIC). Pairs of back-to-back high-transverse-momentum hadrons are used for triggers to study associated hadron distributions. In contrast with two-and three-particle correlations with a single trigger with similar kinematic selections, the associated hadron distribution of both trigger sides reveals no modification in either relative pseudorapidity Delta eta or relative azimuthal angle Delta phi from d + Au to central Au + Au collisions. We determine associated hadron yields and spectra as well as production rates for such correlated back-to-back triggers to gain additional insights on medium properties.

DJKM ALGEBRAS I: THEIR UNIVERSAL CENTRAL EXTENSION

The purpose of this paper is to explicitly describe in terms of generators and relations the universal central extension of the infinite dimensional Lie algebra, g circle times C[t, t(-1), u vertical bar u(2) = (t(2) - b(2))(t(2) - c(2))], appearing in the work of Date, Jimbo, Kashiwara and Miwa in their study of integrable systems arising from the Landau-Lifshitz differential equation.