’t Hooft–Polyakov monopoles and a general spherically symmetric solution of the Bogomolny equations

In this paper, we briefly review the ’t Hooft–Polyakov monopoles and the Bogomolny equations providing minimal energy field configurations in each topologically distinct sectors of monopole solutions. Then we consider the most general spherically symmetric ansatz for gauge fields depending on three arbitrary functions and find a general spherically symmetric solution of the Bogomolny equations. It depends on two integration constants and one arbitrary function on radius. The arbitrary function is proved to be related to the residual gauge symmetry of field configurations. In a general case, the spherically symmetric Euler–Lagrange equations are reduced to two nonlinear field equations for the two invariant functions. The Bogomolny–Prasad–Sommerfield solution is proved to be the only spherically symmetric nonsingular solution of field equations with finite energy.