Small deformations of the Prasad-Sommerfield solution

I study solutions of the static Euclidean anti-self-dual SU(2) Yang-Mills equations which differ by a small perturbation from the Prasad-Sommerfield solution. I find explicit expressions for two series of perturbation mode functions of angular momentum $l$ and even and odd parity, and classify the modes according to several criteria. There are seven nondilatational modes which have singularities removable by gauge transformation: 3 translations ($l=1\mathrm{}$), 1 gauge mode ($l=0\mathrm{}$), and a family of 3 odd-parity gauge modes ($l=1\mathrm{}$). The translations and $l=0\mathrm{}$ gauge modes have nonvanishing, and normalizable, projections into the background gauge, while the odd-parity $l=1\mathrm{}$ modes have vanishing projection into the background gauge. Among the singular modes, there are an infinite number of modes, irregular at $r=0\mathrm{}$, which nonetheless satisfy the boundary conditions for finite-energy solutions on the sphere at infinity. I show, by discussing the analogous problem of the axially symmetric solutions of the stationary Einstein equations, that non-normalizable modes are relevant in determining whether a spherically symmetric solution of a nonlinear system has axially symmetric extensions. The analysis of perturbations around the Prasad-Sommerfield solution implies that if an axially symmetric extension exists, it cannot be reached by integration out along a tangent vector defined by a nonvanishing, nonsingular small-perturbation mode of the class explicitly constructed.