We consider the general classical Heisenberg model (HM) with a z-axis anisotropic Hamiltonian. The ferromagnetic (FR) (antiferromagnetic (AF)) nonlinear spin waves (NLSWs), also called finite-amplitude spin waves, are well-known solutions of the equations of motion and are characterized by constant and equal (in each sublattice, in the AF case) z-components of the spins. In this paper, we present general analytical solutions which share this property, but do not necessarily reside on the equal-spins shell in phase space (spins can be unequal) and hence will be termed 'off-shell' NLSWs. For periodic lattices, we find that these solutions are linear combinations of standard FR (generalized AF) NLSWs. For a Heisenberg ring, in particular, we prove that the 'off-shell' solution is the sum of only two FR (generalized AF) NLSWs of opposite momenta. In this case, we show that the standard NLSWs are the only 'on-shell' solutions with the property that the z-components of the spins (in each sublattice, in the AF case) are all equal to the same nonzero constant. Novel standing-wave solutions with planar spins are also presented.