Asymmetric canonicalization of the integrable nonlinear Schrödinger system on a triangular-lattice ribbon

The integrable nonlinear Schrödinger system on a triangular-lattice ribbon characterized by the essentially nonstandard Poisson structure is shown to be standardized to the nonlinear lattice system consisting of two nonequivalent canonical subsystems. In under-critical region of background parameter both of the canonical subsystems should be treated as the subsystems of bright nonlinear excitations. In the over-critical region one of the canonical subsystem should be treated as the subsystem of bright nonlinear excitations while the another as the subsystem of dark nonlinear excitations.