Eigenfunction equations of lattice KdV equations and connections to ABS lattice equations with a $\delta$ term

We develop lattice eigenfunction equations of lattice KdV equation, which are equations obeyed by the auxiliary functions, or eigenfunctions, of the Lax pair of the lattice KdV equation. This leads to three-dimensionally consistent quad-equations that are closely related to lattice equations in the Adler-Bobenko-Suris (ABS) classification. In particular, we show how the H3($\delta$), Q1($\delta$) and Q3($\delta$) equations in the ABS list arise from the lattice eigenfunction equations by providing a natural interpretation of the $\delta$ term as interactions between the eigenfunctions. By construction, exact solution structures of these equations are obtained. The approach presented in this paper can be used as a systematic means to search for integrable lattice equations.