In this paper, we consider boundary-value and eigenvalue problems for a system of differential equations with variable coefficients, some of which fail to be integrable in any neighborhood of zero. These problems describe vibrations of an inhomogeneous elastic shell.

We introduce certain functional weighted spaces which make it possible to study and consequently to prove the existence and uniqueness of solutions of the boundary-value problem and the existence of solutions of the eigenvalue problem. The properties of the functions of these weighted spaces are also studied, and some embedding theorems are proved.

We would like to thank the referee for many helpful suggestions.