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MIURA TYPE TRANSFORMATIONS AND HOMOGENEOUS SPACES

We relate Miura type transformations (MTs) over an evolution system to its zero-curvature representations with values in Lie algebras . We prove that certain homogeneous spaces of produce MTs and show how to distinguish these spaces. For a scalar translation-invariant evolution equation, this allows us to classify all MTs in terms of homogeneous spaces of the Wahlquist–Estabrook algebra of the equation. For other evolution systems this allows us to construct some MTs. As an example, we study MTs over the KdV equation, a fifth-order equation of Harry Dym type, and the coupled KdV–mKdV system of Kersten and Krasilshchik.