  In this paper we study a special class of fibrations on Delsarte surfaces. We
call these fibrations Delsarte fibrations. We show that after a specific cyclic
base change the fibration is the pull back of a fibration with three singular
fibers, and that this second base change is completely ramified at two points
where the fiber is singular. As a corollary we show that every Delsarte
fibration of genus 1 with nonconstant $j$-invariant occurs as the base change
of an elliptic surface from Fastenberg's list of rational elliptic surfaces
with $\gamma<1$.
