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The relationship between classical and quantum three one-mode systemsinteracting in a non-linear way is described. We investigate the integrabilityof these systems by using the reduction procedure. The reduced coherent statesfor the quantum system are constructed. We find the explicit formulas for thereproducing measure for these states. Examples of some applications of theobtained results in non-linear quantum optics are presented.

Integrability and correspondence of classical and quantum non-linear three-mode system