On Trajectories of Dynamic Systems Lying on Hypersurfaces of Linear Systems

Lagutinskii's theory of integration of dynamic systems is reformulated for arbitrary linear systems of hypersurfaces. The following problems are considered. Some dynamic system and some linear system of algebraic hypersurfaces are given. It is necessary to determine whether any integral curve lies on one of the hypersurfaces of the linear system. In the affirmative case, it is necessary (1) to formulate an equation for this hypersurface and (2) prove the existence of the integral of motion and write an explicit expression for it. An example is constructed showing that the hypersurfaces of the initial linear system and lines of the level of the integral may not coincide.