The paper treats a special type of pseudo-Riemannian spaces, namely those which permit φ(Ric)-vector fields. These spaces are widely applied in mechanics and relativity theory. Authors demonstrated a shape taken by a linear form of basic equations of theory of geodesic mappings for the above-mentioned spaces. These equations take a shape of a system of differential equations in covariant derivatives. Let us find integrability conditions for this system. The research is conducted locally in a tensor form, without limitations on a sign and a signature of a metric tensor.