The uniqueness of a solution to the renewal type system of integral equations on the line

We study the uniqueness of a solution to a renewal type system of integral equations $\text{\mathversion{bold}$z$}=\text{\mathversion{bold}$g$}+\text{\mathversion{bold}$F$}*\text{\mathversion{bold}$z$}$ on the line $\mathbb R$; here {\mathversion{bold}$z$} is the unknown vector function, {\mathversion{bold}$g$} is a known vector function, and {\mathversion{bold}$F$} is a nonlattice matrix of finite measures on $\mathbb R$ such that the matrix $\text{\mathversion{bold}$F$}(\mathbb R)$ is of spectral radius 1 and indecomposable. We show that in a certain class of functions each solution to the corresponding homogeneous system coincides almost everywhere with a right eigenvector of $\text{\mathversion{bold}$F$}(\mathbb R)$ with eigenvalue 1.