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A new set of discrete integrable equations, called face-centered quadequations, was recently obtained using new types of interaction-round-a-facesolutions of the classical Yang-Baxter equation. These equations satisfy a newformulation of multidimensional consistency, known asconsistency-around-a-face-centered-cube (CAFCC), which requires consistency ofan overdetermined system of fourteen five-point equations on the face-centeredcubic unit cell. In this paper a new formulation of CAFCC is introduced whereso-called type-C equations are centered at faces of the face-centered cubicunit cell, whereas previously they were only centered at corners. This allowstype-C equations to be regarded as independent multidimensionally consistentintegrable systems on higher-dimensional lattices and is used to establishtheir Lax pairs.

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