For each irreducible module of the symmetric group \(\mathcal{S}_{N}\) there is a set of parametrized nonsymmetric Jack polynomials in \(N\) variables taking values in the module. These polynomials are simultaneous eigenfunctions of a commutative set of operators, self-adjoint with respect to two Hermitian forms, one called the contravariant form and the other is with respect to a matrix-valued measure on the \(N\)-torus. The latter is valid for the parameter lying in an interval about zero which depends on the module. The author in a previous paper (SIGMA 2016-033) proved the existence of the measure and that its absolutely continuous part satisfies a system of linear differential equations. In this paper the system is analyzed in detail.. The \(N\)-torus is divided into \(\left( N-1\right) !\) connected components by the hyperplanes \(x_{i}=x_{j}\), \(\left( i<j\right) \) which are the singularities of the system. The main result is that the orthogonality measure has no singular part with respect to Haar measure, and thus is given by a matrix function times Haar measure. This function is analytic on each of the connected components.