Constructing SU(2) x U(1) orbit space for qutrit mixed states

The orbit space mathfrak{P}\(mathbb{R}⁸\)/G, of the group G:=SU(2)times U(1)subsetU(3) acting adjointly on the state space mathfrak{P}\(mathbb{R}⁸\) of a 3-level quantum system is discussed. The semi-algebraic structure of mathfrak{P}\(mathbb{R}⁸\) /G, is determined within the Procesi-Schwarz method. Using the integrity basis for the ring of G-invariant polynomials, mathbb{R}\[mathfrak{P}\(mathbb{R}⁸\)\]^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)geqslant 0, is analyzed in details.