Periodic and partially periodic representations of U(IU(n))q are constructed for qm = 1, m being an odd or even integer. The classical non-semisimple IU (n) or U (n)⋉I2n algebra has an abelian subalgebra of dimension 2n. Gelfand-Zetlin bases and matrix elements are generalized and adapted to this case. Our previous results for U(IU(n))q for a generic q (not a root of unity) and those for SU(N)q for qm = 1 are combined in the present study giving explicit matrix elements and eigenvalues such as the second order Casimilar operator D2 = K2 cos(2π/m)(h2n+1+ ... + hnn+1 + n - 1)/cos(2π/m). This displays the role of the internal parameters (hi,n+1) in the q-analogue of the classical K2 (``mass'' squared). The two translation generators (Inn+1, In+1n) become periodic.