Reversible systems with first integrals

A reversible system of ordinary differential equations (ODE) possessing some first integrals is considered. A simple description of the first integral space is given. It is shown that, under the auxiliary assumption that the integrals are symmetric (i.e., they do not change sign under the action of the reversing involution), the presence of the integrals allows one to bring the problem of dynamical stability for a symmetric equilibrium to the similar problem, but for the reduced system. The reduced system is merely the restriction of the initial system to the intersection of the surfaces of level of the integrals passing through the equilibrium.