  We consider harmonic maps on simply connected Riemann surfaces into the group
$\mathrm{U}(n)$ of unitary matrices of order $n$. It is known that a harmonic
map with an associated algebraic extended solution can be deformed into a new
harmonic map that has an $S^1$-invariant associated extended solution. We study
this deformation in detail and show that the corresponding unitons are smooth
functions of the deformation parameter and real analytic along any line through
the origin.
