  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.
