On a generalization of the Lewittes theorem on Weierstrass points

Suppose that $X$ is a compact Riemann surface of genus $g\ge2$, while $\sigma$ is an automorphism of $X$ of order $n$, and $g^*$ is the genus of the quotient surface $X^*=X/\langle\sigma\rangle$. In 1951 Schöneberg obtained a sufficient condition for a fixed point $P\in X$ of $\sigma$ to be a Weierstrass point of $X$. Namely, he showed that $P$ is a Weierstrass point of $X$ if $g^*\ne[g/n]$, where $[x]$ is the integral part of $x$. Somewhat later Lewittes proved the following theorem, equivalent to Schöneberg's theorem: If a nontrivial automorphism $\sigma$ fixes more than four points of $X$ then all of them are Weierstrass points.
These assertions are connected with the notion of a regular covering. We generalize the Lewittes theorem to the case of nonregular coverings and obtain some related corollaries.