Stability Conditions for Solutions of Multidimensional Completely Integrable Differential Equations

New sufficient tests are given for the stability and asymptotic stability of the zero solution of a nonautonomous completely integrable equation on an arbitrary salient convex closed cone and on a finitely generated cone. The class of Lyapunov functions suitable for studying the asymptotic behavior of solutions of nonautonomous completely integrable equations is significantly extended by substantially weakening the sign negativeness condition, traditional in the Lyapunov second method, for the derivative of the Lyapunov function at the interior points of the cone.