On symmetries of singular implicit ODEs

We study implicit ODEs, cubic in derivative, with infinitesimal symmetry at singular points. Cartan showed that even at regular points the existence of nontrivial symmetry imposes restrictions on the ODE. Namely, this algebra has the maximal possible dimension 3 iff the web of solutions is flat. For cubic ODEs with flat 3-web of solutions we establish sufficient conditions for the existence of nontrivial symmetries at singular points and show that under natural assumptions such a symmetry is semi-simple, i.e. is a scaling is some coordinates. We use this symmetry to find first integrals of the ODE.

On the solvability of implicit differential inclusions

In the paper, the set-valued covering mappings are studied. The statements on solvability, solution estimates, and well-posedness of inclusions with conditionally covering mappings are proved. The results obtained are applied to the investigation of differential inclusions unsolved for the unknown function. The statements on solvability, solution estimates, and well-posedness of these inclusions are derived.