  We consider two closely related Riccati equations of constant parameters
whose particular solutions are used to construct the corresponding class of
supersymmetrically-coupled second-order differential equations. We solve
analytically these parametric periodic problems along the positive real axis.
Next, the analytically solved model is used as a case study for a powerful
numerical approach that is employed here for the first time in the
investigation of the energy band structure of periodic not necessarily regular
potentials. The approach is based on the well-known self-matching procedure of
James (1949) and implements the spectral parameter power series solutions
introduced by Kravchenko (2008). We obtain additionally an efficient series
representation of the Hill discriminant based on Kravchenko's series
