On an integrable discretization of integrable systems with a separatrix

An integrable time-discretization of integrable Hamiltonian systems with a separatrix is considered being based on Hirota's bilinear formalism. It is proved that a discrete-time simple pendulum has a complete set of exact solutions, a conserved quantity and a separatrix. The value of the conserved quantity which characterizes the separatrix is remarkably congruent with the value of the continuous-time Hamiltonian. A discrete-time anharmonic oscillator is also shown to have the same property.