Commutativity of Pfaffianization and Bäcklund transformations for a special two-dimensional lattice

In this Letter, we first present the Wronskian solutions to the Bäcklund transformation of a special two-dimensional lattice by Blaszak and Szum [M. Blaszak, A. Szum, J. Math. Phys. 42 (2001) 225] and then derive the coupled system for the Bäcklund transformation through Pfaffianization. It is shown the coupled system so obtained is nothing but the Bäcklund transformation for the coupled system of the special lattice [G.F. Yu, C.X. Li, J.X. Zhao, X.B. Hu, J. Nonlinear Math. Phys. 12 (2005) 316]. This implies that Pfaffianization and Bäcklund transformations are commutative for the special lattice. Moreover, since the two-dimensional Toda lattice equation forms one part of the special lattice, it is obvious that the commutativity is also applicable to it.