The authors derive a 3-parameter family of well posed versions of the conformally-decomposed \(3+1\) Einstein equations, perhaps amenable to successful numerical integration. This well posed version requires the lapse to be proportional to the determinant of the intrinsic geometry of the surfaces, and requires combinations of the constraints with the evolution equations. The lapse density and the shift vector are arbitrary nondynamical variables, which means that they must be specified as free source functions. The characteristics structure is computed, and the constraint propagation of new well posed formulations is verified.