A criterion for a Poisson matrix determinant to be a partial integral of the Hamiltonian system

Consider a Hamiltonian system, restricted onto an invariant surface. Does it have an integral, which may be explicitly expressed through the equations, determining this submanifold? A simple criterion of the existence of partial integral, equal to their Poisson matrix determinant, has been found. This integral is not trivial iff the induced Poisson structure is nondegenerate at least at one point. Particularly, the submanifold is to be even-dimensional.