[Dedicated to the memory of Sergei Manakov]
Ermakov-type invariants are isolated for a subsystem of an N-component coupled nonlinear Schrödinger system. An algorithmic procedure is presented which reduces this Ermakov–Ray–Reid system to quadrature. The method is illustrated in the single component case by application to a nonlinear system descriptive of the propagation of transverse waves in incompressible hyperelastic media subject to rotation. An extended Ermakov–Ray–Reid system is presented which, if it has underlying Hamiltonian structure, is also amenable to the algorithm.