Let k be a perfect field with algebraic closure k. If H is a subgroup of plane automorphisms over k and p ∈ k² is a point, we describe the subgroup consisting of plane automorphisms which stabilize the orbit of p under H, when this orbit has irreducible closure in k². As an application, we treat the case where H is cyclic and the closure of the orbit of p is an arbitrary (non-necessarily irreducible) curve.

Research of both authors was partially supported by CSIC (Udelar), ANII and PEDECIBA, of Uruguay.