The full automorphism group of a cyclic p-gonal surface

If p is prime, a compact Riemann surface X of genus g⩾2 is called cyclic p-gonal if it admits a cyclic group of automorphisms Cp of order p such that the quotient space X/Cp has genus 0. If in addition Cp is not normal in the full automorphism A, then we call X a non-normal p-gonal surface. In the following we classify all non-normal p-gonal surfaces.