"Quantization" of higher hamiltonian analogues of the Painleve I and Painleve II equations with two degrees of freedom

We construct a solution of an analog of the Schr"{o}dinger equation for the Hamiltonian $H_I(z,t,q_1,q_2,p_1,p_2)$ corresponding to the second equation $P_1^2$ in the Painleve I hierarchy. This solution is produced by an explicit change of variables from a solution of the linear equations whose compatibility condition is the ordinary differential equation $P_1^2$ with respect to $z$. This solution also satisfies an analog of the Schr"{o}dinger equation corresponding to the Hamiltonian $H_{II}(z,t,q_1,q_2,p_1,p_2)$ of Hamiltonian system with respect to $t$ which is compatible with $P_1^2$. A similar situation occurs for the $P_2^2$ equation in the Painleve II hierarchy.