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

The orbit space $\mathfrak{P}({\mathbb{R}}^8)\mathrm{/}\mathrm{G}$, of the group $\mathrm{G}:=\mathrm{S}\mathrm{U}(2)\times \mathrm{U}(1)\subset \mathrm{U}(3)$ 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, $\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{d}(z)⩾0$, is analyzed in details.