Classical integrable field theories in discrete (2+1) -dimensional spacetime

We study the 'circular net' (discrete orthogonal net) equations for the angular data generalized by external spectral parameters. A criterion defining physical regimes of these Hamiltonian equations is the reality of the Lagrangian density. There are four distinct regimes for fields and spectral parameters classified by four types of spherical or hyperbolic triangles. Nonzero external spectral parameters provide the existence of field-theoretical ground states and soliton excitations. Spectral parameters of a spherical triangle correspond to a statistical mechanics; spectral parameters of hyperbolic triangles correspond to three different field theories with massless anisotropic dispersion relations.