Generalizations of the Camassa–Holm equation

We classify generalized Camassa–Holm-type equations which possess infinite hierarchies of higher symmetries. We show that the obtained equations can be treated as negative flows of integrable quasi-linear scalar evolution equations of orders 2, 3 and 5. We present the corresponding Lax representations or linearization transformations for these equations. Some of the obtained equations seem to be new.

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