  We propose the mapping of polynomial of degree 2S constructed as a linear
combination of powers of spin-$S$ (for simplicity, we called as spin-$S$
polynomial) onto spin-crossover state. The spin-$S$ polynomial in general can
be projected onto non-symmetric degenerated spin up (high-spin) and spin down
(low-spin) momenta. The total number of mapping for each general spin-$S$ is
given by $2(2^{2S}-1)$. As an application of this mapping, we consider a
general non-bilinear spin-$S$ Ising model which can be transformed onto
spin-crossover described by Wajnflasz model. Using a further transformation we
obtain the partition function of the effective spin-1/2 Ising model, making a
suitable mapping the non-symmetric contribution leads us to a spin-1/2 Ising
model with a fixed external magnetic field, which in general cannot be solved
exactly. However, for a particular case of non-bilinear spin-$S$ Ising model
could become equivalent to an exactly solvable Ising model. The transformed
Ising model exhibits a residual entropy, then it should be understood also as a
frustrated spin model, due to competing parameters coupling of the non-bilinear
spin-$S$ Ising model.
