  We consider some classical and frustrated lattice spin models with global
O(3) spin symmetry. There is no general analytical method to find a
ground-state if the spin dependence of the Hamiltonian is more than quadratic
(i.e. beyond the Heisenberg model) and/or if the lattice has more than one site
per unit cell. To deal with these situations, we introduce a family of
variational spin configurations, dubbed "regular states", which respect all the
lattice symmetries modulo global O(3) spin transformations (rotations and/or
spin flips). The construction of these states is explicited through a group
theoretical approach, and all the regular states on the square, triangular,
honeycomb and kagome lattices are listed. Their equal time structure factors
and powder-averages are shown for comparison with experiments. All the well
known N\'eel states with 2 or 3 sublattices appear amongst regular states on
various lattices, but the regular states also encompass exotic non-planar
states with cubic, tetrahedral or cuboctahedral geometry of the T=0 order
parameter. Whatever the details of the Hamiltonian (with the same symmetry
group), a large fraction of these regular states are energetically stationary
with respect to small deviations of the spins. In fact these regular states
appear as exact ground-states in a very large range of parameter space of the
simplest models that we have been looking at. As examples, we display the
variational phase diagrams of the J1-J2-J3 Heisenberg model on all the previous
lattices as well as that of the J1-J2-K ring-exchange model on square and
triangular lattices.
