  We establish and analyze a new relationship between the matrices describing
an arbitrary component of a spin $s$, where $2s\in \mathbb{Z}^+$, and the
matrices of $\mathbb{C}P^{2s}$ two-dimensional Euclidean sigma models. The spin
matrices are constructed from the rank-1 Hermitian projectors of the sigma
models or from the antihermitian immersion functions of their soliton surfaces
in the $\mathfrak{su}(2s+1)$ algebra. For the spin matrices which can be
represented as a linear combination of the generalized Pauli matrices, we find
the dynamics equation satisfied by its coefficients. The equation proves to be
identical to the stationary equation of a two-dimensional Heisenberg model. We
show that the same holds for the matrices congruent to the generalized Pauli
ones by any coordinate-independent unitary linear transformation. These
properties open the possibility for new interpretations of the spins and also
for application of the methods known from the theory of sigma models to the
situations described by the Heisenberg model, from statistical mechanics to
quantum computing.
