  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 to
classify all MTs in terms of homogeneous spaces of the Wahlquist-Estabrook
algebra of the equation. For other evolution systems this allows to construct
some MTs. As an example, we study MTs over the KdV equation, a 5th order
equation of Harry-Dym type, and the coupled KdV-mKdV system of Kersten and
Krasilshchik.
