Lax form of the quantum mechanical eigenvalue problem

The quantum mechanical eigenvalue problem with the hamiltonian H = H0 + tV is written as a set of dynamical equations for the eigenvalues xn(t) and the matrix elements Vnm(t) regarding the parameter t as time. By appropriate changes of variables it can be expressed as a pair of matrix equations with the Lax form, hence we are able to write all the possible constants of the motion explicitly. Implications of these constants to the statistical properties of levels are discussed