“Quantizations” of higher Hamiltonian analogues of the Painlevé I and Painlevé II equations with two degrees of freedom

We construct a solution of an analogue of the Schrödinger equation for the Hamiltonian H1(z, t, q1, q2, p1, p2) corresponding to the second equation P12 in the Painlevé I hierarchy. This solution is obtained by an explicit change of variables from a solution of systems of linear equations whose compatibility condition is the ordinary differential equation P12 with respect to z. This solution also satisfies an analogue of the Schrödinger equation corresponding to the Hamiltonian H2(z, t, q1, q2, p1, p2) of a Hamiltonian system with respect to t compatible with P12. A similar situation occurs for the P22 equation in the Painlevé II hierarchy.