On meromorphic solutions of the equations related to the non-stationary hierarchy of the second Painleve equation

The non-stationary hierarchy of the second Painleve equation is herein considered. It is a sequence of polynomial ordinary differential equations of even order with a single differential-algebraic structure determined by the operator $\tilde{L}_{N}$. The first member of this hierarchy for $N = 1$ is the second Painleve equation, and the subsequent equations of $2N$ order contain arbitrary parameters. They are also named generalised higher analogues of the second Painleve equation of $2N$ order. The hierarchies of the first Painleve equation and the equation $P_{34}$ from the classification list of canonical Painleve equations are also associated with this hierarchy. In this paper, we also consider a second order linear equation the coefficients of which are determined by solutions of the hierarchy of the second Painleve equation and the equation $P_{34}$. Using the Frobenius method, we obtain sufficient conditions for the meromorphicity of the general solution of second-order linear equations with the coefficients defined by the solutions of the first three equations of the non-stationary hierarchy of the second Painleve equation and the equation $P_{34}$. We also find sufficient conditions for the rationality of the general solution of second-order linear equations with coefficients determined by rational solutions of the equations of the non-stationary hierarchy of the second Painleve equation and the equation $P_{34}$.