On a Partial Integral which can be Derived from Poisson Matrix

Consider a surface which is a common level of some functions. Suppose that this surface is invariant under a Hamiltonian system. The question is if a partial integral can be derived explicitly from the Poisson matrix of these functions. In some cases such an integral is equal to the determinant of the matrix. This paper establishes a necessary and sufficient condition for this to hold true. The partial integral that results is not trivial if the induced Poisson structure is non-degenerate at one point at least. Therefore, the invariant surface must be even-dimensional.