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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 onthe state space $\mathfrak{P}(\mathbb{R}^8)$ of a 3-level quantum system isdiscussed. The semi-algebraic structure of $\mathfrak{P}(\mathbb{R}^8)/\mathrm{G}$, is determined within the Procesi-Schwarz method. Using theintegrity basis for the ring of G-invariant polynomials,$\mathbb{R}[\mathfrak{P}(\mathbb{R}^8)]^{\mathrm{G}}$, the set of constraintson the Casimir invariants of $\mathrm{U}(3)$ group coming from the positivityrequirement of Procesi-Schwarz gradient matrix, $\mathrm{Grad}(z)\geqslant 0$,is analyzed in details.

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