  Let X be an affine irreducible variety over an algebraically closed field k
of characteristic zero. Given an automorphism F, we denote by k(X)^F its field
of invariants, i.e. the set of rational functions f on X such that f(F)=f. Let
n(F) be the transcendence degree of k(X)^F over k. In this paper, we study the
class of automorphisms F of X for which n(F)= dim X - 1. More precisely, we
show that under some conditions on X, every such automorphism is of the form
F=A_g, where A is an algebraic action of a linear algebraic group G of
dimension 1 on X, and where g belongs to G. As an application, we determine the
conjugacy classes of automorphisms of the plane for which n(F)=1.
