  We study the BPS condition in the $\Omega$-deformed $\mathcal{N}=2$ super
Yang-Mills theory when one of the $\epsilon$-parameters of the background is
zero. We obtain the deformed BPS equation for dyons and the formula for their
central charge. In particular, we find that the deformed BPS monopole equation
has axially-symmetric solution and is equivalent to the Ernst equation. The
monopole charge is shown to be undeformed. We construct one-monopole solution
explicitly and examine its profile.
