A Revisit to the ABS $\mathrm{H2}$ Equation

In this paper we revisit the Adler–Bobenko–Suris $\mathrm{H2}$ equation. The $\mathrm{H2}$ equation is linearly related to the $S^{(0,0)}$ and $S^{(1,0)}$ variables in the Cauchy matrix scheme. We elaborate the coupled quad-system of $S^{(0,0)}$ and $S^{(1,0)}$ in terms of their $3$-dimensional consistency, Lax pair, bilinear form and continuum limits. It is shown that $S^{(1,0)}$ itself satisfies a $9$-point lattice equation and in continuum limit $S^{(1,0)}$ is related to the eigenfunction in the Lax pair of the Korteweg–de Vries equation.