  In this paper we show how to bypass the usual difficulties in the analysis of
elliptic integrals that arise when solving period problems for minimal
surfaces. The method consists of replacing period problems with ordinary
Sturm-Liouville problems involving the support function. We give a practical
application by proving existence of the sheared Scherk-Karcher family of
surfaces numerically described by Wei. Moreover, we show that this family is
continuous, and both of its limit-members are the singly periodic genus-one
helicoid.
