On consistency around a $3 \times 3\times 3$ cube and Q3 analogue of the lattice Boussinesq equation

In this paper, we present two new aspects of lattice Boussinesq (BSQ) equations. First, we show that the lattice potential BSQ (lpBSQ) equation defined on a nine-point square lattice admits a natural extension of three-dimensional consistency to a $3\times 3\times 3$ cube\textemdash a cubic sublattice consisting of $27$ vertices. This extends the standard notion of three-dimensional consistency (defined on an elementary $2\times 2\times 2$ vertex cube for quadrilateral equations) to the non-quadrilateral, nine-point setting. Second, we construct a new three-component system which is referred to as the {\em lattice BSQ-Q3 system}, serving as the BSQ analogue of the Q3($δ$) equation in the Adler-Bobenko-Suris (ABS) classification. The construction relies on a gauge transformation between Lax pairs of lpBSQ with the parameter $δ$ arising from a $GL_3$ action. In a degeneration form, the system yields a $PGL_3$-invariant integrable lattice equation that generalises the $PGL_2$-invariant Schwarzian BSQ equation.