In a recent study Akarsu and Dereli (Gen. Relativ. Gravit. 45:1211, 2013) discussed the dynamical reduction of a higher dimensional cosmological model which is augmented by a kinematical constraint characterized by a single real parameter, correlating and controlling the expansion of both the external (physical) and internal spaces. In that paper explicit solutions were found only for the case of three dimensional internal space (n = 3). Here we derive a general solution of the system using Lie group symmetry properties, in parametric form for arbitrary number n = 1, 2, 3, ... of internal dimensions. We also investigate the dynamical reduction of the model as a function of cosmic time t for various values of n and generate parametric plots to discuss cosmologically relevant results.