  Integrable difference equations commonly have more low-order conservation
laws than occur for nonintegrable difference equations of similar complexity.
We use this empirical observation to sift a large class of difference
equations, in order to find candidates for integrability. It turns out that all
such candidates have an equivalent affine form. These are tested by calculating
their algebraic entropy. In this way, we have found several types of integrable
equations, one of which seems to be entirely unrelated to any known discrete
integrable system. We also list all single-tile conservation laws for the
integrable equations in the above class.
