Equivariant deformations of LeBrun's self-dual metrics with torus action

We investigate -equivariant deformations of C. LeBrun's self-dual metric with torus action. We explicitly determine all -subgroups of the torus for which one can obtain -equivariant deformations that do not preserve the whole of the torus action. This gives many new self-dual metrics with -action which are not conformally isometric to LeBrun metrics. We also count the dimension of the moduli space of self-dual metrics with -action obtained in this way.