"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 an-alytically 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 thefirst time in the investigation of the en-ergy band structure of periodic not necessarily regular potentials. The approach is based on the well-known self-matching procedure of James (1949) and imple-ments the spectral parameter power series solutions introduced by Kravchenko (2008). We obtain additionally an efficient series representation of the Hill dis-criminant based on on Kravchenko´s series."