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 g. We prove that certain homogeneous spaces of g 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