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

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