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The article is devoted to the description of dynamics of magnets witharbitrary spin on the basis of the Hamiltonian formalism. The relationshipbetween the magnetic ordering and Poisson bracket subalgebras of the magneticdegrees of freedom for spin s=1/2; 1; 3/2 has been established. We have beenobtained non-linear dynamic equations without damping for normal and degeneratenon-equilibrium states of high-spin magnets with the properties of the SO(3),SU(4), SU(2)$\times$SU(2), SU(3), SO(4), SO(5) symmetry of exchangeinteraction. The connection between models of the magnetic exchange energy andthe Casimir invariants has been discussed.

SU(2s+1) symmetry and nonlinear dynamic equations of spin s magnets