An infinite set of the operator-valued relations in the Liouville field theory is established. These relations are enumerated by a pair of positive integers (m, n), the first (1,1) representative being the usual Liouville equation of motion. The relations are proven in the framework of conformal field theory on a basis of the exact structure constants in the Liouville operator product expansions. Possible applications in 2D gravity are discussed.
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Al.B. Zamolodchikov (2004) studied this question.