Discrete integrable equations on face-centered cubics: consistency and Lax pairs of corner equations

A new set of discrete integrable equations, called face-centered quad equations, was recently obtained using new types of interaction-round-a-face solutions of the classical Yang-Baxter equation. These equations satisfy a new formulation of multidimensional consistency, known as consistency-around-a-face-centered-cube (CAFCC), which requires consistency of an overdetermined system of fourteen five-point equations on the face-centered cubic unit cell. In this paper a new formulation of CAFCC is introduced where so-called type-C equations are centered at faces of the face-centered cubic unit cell, whereas previously they were only centered at corners. This allows type-C equations to be regarded as independent multidimensionally consistent integrable systems on higher-dimensional lattices and is used to establish their Lax pairs.