Calogero-Moser Spaces and the Invariants of Two Matrices of Degree 3

We find a minimal set of generators for the coordinate ring Calogero-Moser space ${\boldsymbol {\mathcal {C}}}_{\boldsymbol {3}}$ and algebraic relations among them explicitly. give new presentation algebra 3×3 invariant matrices involving defining ${\mathbb {C}}[\boldsymbol {C}}_{\boldsymbol {3}}]$ . Furthermore, we an explicit description GL3-invariant commuting variety its orbits under action affine plane Cremona group.