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

We investigate U(1)-equivariant deformations of C. LeBrun's self-dual metric with torus action. We explicitly determine all U(1)-subgroups of the torus for which one can obtain U(1)-equivariant deformation that do not preserve semi-free U(1)-action. This gives many new self-dual metrics with U(1)-action which are not conformally isometric to LeBrun metric. We also count the dimension of the moduli space of self-dual metrics with U(1)-action obtained in this way.