  We study deformations of axially symmetric initial data for Einstein-Maxwell
equations satisfying the time-rotation ($t$-$\phi$) symmetry and containing one
asymptotically cylindrical end and one asymptotically flat end. We find that
the $t$-$\phi$ symmetry implies the existence of a family of deformed data
having the same horizon structure. This result allows us to measure how close
solutions to Lichnerowicz equation are when arising from nearby free data.
