episciences.org_9798_1665089723 1665089723 episciences.org raphael.tournoy+crossrefapi@ccsd.cnrs.fr episciences.org Open Communications in Nonlinear Mathematical Physics 2802-9356 10.46298/journals/ocnmp https://ocnmp.episciences.org 08 02 2022 Volume 2 Search for integrable two-component versions of the lattice equations in the ABS-list Jarmo Hietarinta We search and classify two-component versions of the quad equations in the ABS list, under certain assumptions. The independent variables will be called \$y,z\$ and in addition to multilinearity and irreducibility the equation pair is required to have the following specific properties: (1) The two equations forming the pair are related by \$y\leftrightarrow z\$ exchange. (2) When \$z=y\$ both equations reduce to one of the equations in the ABS list. (3) Evolution in any corner direction is by a multilinear equation pair. One straightforward way to construct such two-component pairs is by taking some particular equation in the ABS list (in terms of \$y\$), using replacement \$y \leftrightarrow z\$ for some particular shifts, after which the other equation of the pair is obtained by property (1). This way we can get 8 pairs for each starting equation. One of our main results is that due to condition (3) this is in fact complete for H1, H3, Q1, Q3. (For H2 we have a further case, Q2, Q4 we did not check.) As for the CAC integrability test, for each choice of the bottom equations we could in principle have \$8^2\$ possible side-equations. However, we find that only equations constructed with an even number of \$y \leftrightarrow z\$ replacements are possible, and for each such equation there are two sets of "side" equation pairs that produce (the same) genuine B\"acklund transformation and Lax pair. 08 02 2022 9798 arXiv:2009.12208 10.48550/arXiv.2009.12208 https://arxiv.org/abs/2009.12208v2 10.46298/ocnmp.9798 https://ocnmp.episciences.org/9798 https://ocnmp.episciences.org/9881/pdf