  We prove that the critical points of various energies such as the area, the
Willmore energy, the frame energy for tori...etc among possibly branched
immersions constrained to evolve within a smooth sub-manifold of the
Teichm\"uller space satisfy the corresponding constrained Euler Lagrange
equation. We deduce that critical points of the Willmore energy or the frame
energy for tori are smooth analytic surfaces, away possibly from isolated
branched points, under the condition that either the genus is at most 2 or if
the sub-manifold does not intersect the subspace of hyper-elliptic points.
Using a compactness result from a previous work of the author, we can conclude
that each closed sub-manifold of the Teichm\"uller space, including points,
under the previous assumptions, posses a possibly branched smooth Willmore
minimizer satisfying the conformal-constrained Willmore equation.
