Nonstandard separation of variables for the Maxwell–Bloch conservative system

he Maxwell–Bloch conservative system, which originated in laser optic theory, is a nice, well-known example of a completely integrable mechanical system. Here we show how to separate the system into two subsystems: one with a fish-shaped dynamics and the other which is governed by a central force. The behaviour of the whole system is then easily explained with tools that are more elementary than in previous approaches. The separation of the fish-shaped dynamics is new, and it is obtained in an unconventional way, using nonlocal constants of motion.