An explicit solution is developed of the matrix equation A(z)X(z) + X(z)B(z) = C(z) where z is a parameter. The approach considered is general in that no particular representations are assumed and the solution X(z) is expressed as a simple function of the matrices involved. As an application, the stability of a class of generalized linear systems is discussed, namely systems which depend polynomially on a real parameter and time-delay systems.