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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, separableterms in each entry of each Lax matrix, along with the most general nonlinearsystems that can be associated with them. Here we allow some of the termswithin the Lax matrices to be zero. We cover all possible Lax pairs of thistype and find a new third order equation that can be reduced to special casesof the non-autonomous lattice KdV and lattice modified KdV equations amongothers.

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