Constructing $\mathrm{SU(2)\times U(1)}$ orbit space for qutrit mixed states

The orbit space $\mathfrak P(\mathbb R^8)/\mathrm G$ of the group
$$ \mathrm{G:=SU(2)\times U(1)\subset U(3)} $$
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 $\mathrm G$-invariant polynomials $\mathbb R[\mathfrak P(\mathbb R^8)]^\mathrm G$, the set of constraints on the Casimir invariants of the group $\mathrm U(3)$ coming from the positivity requirement for Procesi–Schwarz gradient matrix, $\mathrm{Grad}(z)\geqslant0$, is analyzed in detail.