On integrability of discrete variational systems. Octahedron relations
  We elucidate consistency of the so-called corner equations which are elementary building blocks of Euler-Lagrange equations for two-dimensional pluri-Lagrangian problems. We show that their consistency can be derived from the existence of two independent octahedron relations. We give explicit formulas for octahedron relations in terms of corner equations. 
Raphael Boll, Matteo Petrera, Yuri B. Suris
nlin.SI math-ph math.MP