The “quantum” linearization of the Painlevé equations as a component of theier $L,A$ pairs

The procedure of the “quantum” linearization of the Hamiltonian ordinary differential equations with one degree of freedom is investigated. It is offered to be used for the classification of integrable equations of the Painleve type. For the Hamiltonian $H=(p^2+q^2)/2$ and all natural numbers $n$ the new solutions $\Psi(\hbar,t,x,n)$ of the non-stationary Shrödinger equation are constructed. The solutions tend to zero at $x\to\pm\infty$. On curves $x=q_n(\hbar,t)$, defined by the old Bohr–Zommerfeld rule, the solutions satisfy the relation $i\hbar\Psi'_x\equiv p_n(\hbar,t)\Psi$. In this relation $p_n(\hbar,t)=(q_n(\hbar,t))'_t $ is the classical momentum corresponding to the harmonic $q_n(\hbar,t)$.