Abstract This paper deals with a problem of finding a bounded in a strip solution to a system of second order hyperbolic evolution equations, where the matrix coefficient of the spatial derivative tends to zero as t → ∓ ∞ . The problem is studied under assumption that the coefficients, the right-hand side of the system, and the boundary function belong to some spaces of functions continuous and bounded with a weight. By introducing new unknown functions, the problem in question is reduced to an equivalent problem consisting of singular boundary value problems for a family of first order ordinary differential equations and some integral relations. Existence conditions are established for a bounded in a strip solutions to a family of ordinary differential equations, whose matrix tends to zero as t → ∓ ∞ and the right-hand side is bounded with a weight. Conditions for the existence of a unique solution to the original problem are obtained.