  In this article we study various analytic aspects of interpolating
sesqui-harmonic maps between Riemannian manifolds where we mostly focus on the
case of a spherical target. The latter are critical points of an energy
functional that interpolates between the functionals for harmonic and
biharmonic maps. In the case of a spherical target we will derive a
conservation law and use it to show the smoothness of weak solutions. Moreover,
we will obtain several classification results for interpolating sesqui-harmonic
maps.
