On meromorphic solutions of the equations related to the first Painlev$\acute{e}$ equation

In this paper, we consider the generalised hierarchy of the first Painlev$\acute{e}$ equation which is a sequence of polynomial ordinary differential equations of even order that have a uniform differential-algebraic structure determined by the operator $\tilde{L_{n}}$. The first member of this hierarchy for $n=2$ is the first Painlev$\acute{e}$ equation, and the subsequent equations of order $2n-2$ contain arbitrary parameters. They are named as higher analogues of the first Painlev$\acute{e}$ equation of $2n-2$ order. The article considers the analytical properties of solutions to the equations of the generalised hierarchy of the first Painlev$\acute{e}$ equation and the related linear equations. It is established that each hierarchy equation has one dominant term, and an arbitrary meromorphic solution of any hierarchy equation cannot have a finite number of poles. The character of the mobile poles of meromorphic solutions is determined. Using the Frobenius method, sufficient conditions are obtained for the meromorphicity of the general solution of the second-order linear equations with a linear potential defined by meromorphic solutions of the first three equations of the hierarchy.