Infinitely many Riemann surfaces with transitive action on Weierstrass points

In this short note, we prove the existence of infinitely many pairwise non-isomorphic non-hyperelliptic Riemann surfaces with automorphism group acting transitively on the Weierstrass points. We also found all those compact Riemann surfaces with automorphism group acting transitively on the Weierstrass points, under the assumption that they are simple.