  The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a
proper generalization to 3 dimensions of the zero curvature relation. Recently,
Korepanov has constructed an infinite set of integrable 3-dimensional lattice
models, and has related them to solutions to the local YBE. The simplest
Korepanov's model is related to the star-triangle relation in the Ising model.
In this paper the corresponding discrete equation is derived. In the continuous
limit it leads to a differential 3d equation, which is symmetric with respect
to all permutations of the three coordinates. A similar analysis of the
star-triangle transformation in electric networks leads to the discrete
bilinear equation of Miwa, associated with the BKP hierarchy. Some related
operator solutions to the tetrahedron equation are also constructed.
