Abstract: The centralizer of a diffeomorphism f is the set of diffeomorphisms g that commute with f. In other words, the centralizer of f is the group of symmetries of f, where "symmetries" is meant the classical sense: coordinate changes leave the dynamics of the system unchanged. For certain algebraic examples, they may have exceptional large symmetry group, i.e. the centralizer contains a non-trivial Lie group. For example, the centralizer of the time-1 map f_0 of the geodesic flow of a negatively curved surface contains R, etc. A natural question is: how about the symmetry group of a perturbation of f_0? This relates to one of the classical questions in perturbation theory: if a diffeomorphism belongs to a smooth flow, which perturbations also belong to a smooth flow? In this talk we will fully classify the centralizer of ALL (not only generic) C^1 conservative perturbation of f_0 above. And this is an example of the joint work with D. Damjanovic and A. Wilkinson.