We investigate the light-cone SU(n) Yang-Mills mechanics formulated as the leading order of the longwavelength approximation to the light-front SU(n) Yang-Mills theory. In the framework of the Dirac formalism for degenerate Hamiltonian systems, for models with the structure groups SU(2) and SU (3), we determine the complete set of constraints and classify them. We show that the light-cone mechanics has an extended invariance: in addition to the local SU(n) gauge rotations, there is a new local twoparameter Abelian transformation, not related to the isotopic group, that leaves the Lagrangian system unchanged. This extended invariance has one profound consequence. It turns out that the light-cone SU(2) Yang-Mills mechanics, in contrast to the well-known instant-time SU(2) Yang-Mills mechanics, represents a classically integrable system. For calculations, we use the technique of Gröbner bases in the theory of polynomial ideals.

Entangling properties of a mixed 2-qubit system can be described by the local homogeneous unitary invariant polynomials in elements of the density matrix. The structure of the corresponding invariant polynomial ring for the special subclass of states, the so-called mixed X−states, is established. It is shown that for the X−states there is an injective ring homomorphism of the quotient ring of SU (2)×SU (2) invariant polynomials modulo its syzygy ideal and the SO(2) × SO(2)−invariant ring freely generated by five homogeneous polynomials of degrees

The orbit space P(R 8 )/G of the group G := SU(2) × U(1) ⊂ U(3) acting adjointly on the state space P(R 8 ) of a 3-level quantum system is discussed. The semi-algebraic structure of P(R 8 )/G is determined within the Procesi-Schwarz method. Using the integrity basis for the ring of G-invariant polynomials, R[P(R 8 )] G , the set of constraints on the Casimir invariants of U(3) group coming from the positivity requirement of Procesi-Schwarz gradient matrix, Grad(z) 0 , is analyzed in details.