A Completeness Study on Certain $2\times2$ Lax Pairs Including Zero Terms

TL;DR

This paper extends previous classifications of 2x2 Lax pairs by including zero terms, identifying all such pairs and discovering a new third order equation related to known integrable lattice equations.

Contribution

It provides a comprehensive classification of 2x2 Lax pairs with zero entries and introduces a new third order integrable equation.

Findings

Classified all 2x2 Lax pairs with zero terms

Discovered a new third order integrable equation

Connected the new equation to known lattice KdV variants

Abstract

We expand the completeness study instigated in [J. Math. Phys. 50 (2009), 103516, 29 pages] which found all $2\times2$ Lax pairs with non-zero, separable terms in each entry of each Lax matrix, along with the most general nonlinear systems that can be associated with them. Here we allow some of the terms within the Lax matrices to be zero. We cover all possible Lax pairs of this type and find a new third order equation that can be reduced to special cases of the non-autonomous lattice KdV and lattice modified KdV equations among others.