Continuous deformations of harmonic maps and their unitons

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.