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 deformations that do not preserve the whole of the torus action. This gives many new self-dual metrics with $U(1)$-action which are not conformally isometric to LeBrun metrics. We also count the dimension of the moduli space of self-dual metrics with $U(1)$-action obtained in this way.