  We prove that the conformal immersions of complex two tori into $S^3$ which
locally minimize their conformal volume in their conformal class all satisfy
some elliptic PDE. We prove that they are either minimal tori, CMC flat tori,
elliptic conformally constrained minimal tori or critical point of the area
under some fixed conformally congruent area. On the way to establish this
result we prove that tori which are critical points of the area for
perturbations within a given conformal class and which are degenerate points of
the conformal class mapping - i.e. isothermic - are either minimal surfaces or
flat CMC tori. These results are all proved in the general framework of weak
immersions.
