Many-Dimensional Poincaré Construction and Singularities of Lifted Fields For Implicit Differential Equations

The paper is devoted to singular points of the so-called lifted vector fields, which arise in studying systems of implicit differential equations by using the method of lifting the equation to a surface, a generalization of the construction used by Poincaré for a single implicit equation. The author study the phase portraits and renormal forms of such fields in a neighborhood of their singular points. In conclusion, the paper considers the lifted vectors fields generated by Euler–Lagrange and Euler–Poisson equations and fast-slow systems.