Sufficient commuting integrals for odd-dimensional inhomogeneous [j,k,0] systems with a specific inhomogeneous term

Ascertain whether the odd-dimensional inhomogeneous Lotka–Volterra system of type [j,k,0] in dimension n = 2(j + k) + 1, with inhomogeneous term r_i = r for i < n and r_n = r (a_{j+3k−1} − a_{j+3k})/a_{j+3k−2}, admits a set of commuting, functionally independent first integrals sufficient for Liouville integrability; equivalently, construct such integrals or prove that they do not exist.

Background

In extending their even-dimensional constructions to odd dimensions, the authors consider inhomogeneous terms of the form r = (r,…,r,s) that preserve certain homogeneous weight-zero integrals. For the general odd [j,k,l] case with the proposed inhomogeneity, they show the resulting set of integrals is insufficient for Liouville integrability.

Specializing to the odd [j,k,0] case with a particular inhomogeneous term, the authors explicitly state they have not been able to construct enough commuting integrals. Determining whether sufficient commuting integrals exist (or proving their nonexistence) would settle the Liouville integrability of this class under that inhomogeneity.