Complete sets of solutions of linear Lorentz covariant field equations with an infinite number of field components

We study field equations of the Gelfand-Yaglom type where transforms as a unitary representation of the inhomogeneous Lorentz group. We construct a complete set of solutions of this equation. This set includes solutions with spacelike momentum. Our method makes use of the decomposition of unitary representations of the homogeneous Lorentz group into unitary representations of the little groupsS U (2) andS U (1, 1). The covariant operators ^µ are written as differential operators on homogeneous spaces. For some classes of equations we calculate the mass spectrum explicitly.