  We study "circular net" (discrete orthogonal net) equations for angular data
generalized by external spectral parameters. A criterion defining physical
regimes of these Hamiltonian equations is the reality of Lagrangian density.
There are four distinct regimes for fields and spectral parameters classified
by four types of spherical or hyperbolic triangles. Non-zero 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.
