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Ergodic theory is constructed by fixing the dynamics on a surface of the phase space with constant energy. In case a non-integrable system conserved more additional quantities apart from the energy, is it possible to construct a 'generalized ergodic theory' on the lower-dimensional hyper-surface of the phase space which, accounting for the additional conservation laws, is still metrically indecomposible w.r.t. dynamics?

Any reference where such topic is discussed would be also really appreciated.

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