The concept of non-Néel order parameter (double, triple, etc. spin correlators as in the exchange approximation) is generalized to the strongly anisotropic magnets. As the coherent wave functions of the spins we consider various degenerate representations for the multiplets. We consider then the coherent wave function from the symmetry point of view. For the atomic wave functions of local moments split by the crystal field multiplets with integer and half-integer spins have different properties. We also address the issue of small staggered moments in URu 2Si 2 and UPt 3 at low temperatures.