Differential substitutions of the Miura transformation type

The existence of a nontrivial integral and a sufficiently large set of symmetries with respect to one of the characteristics of a hyperbolic equation implies the existence of nontrivial integrals and symmetries with respect to the other characteristic. We classify differential substitutions of the form $w=P(x,u,u_{x})$ relating families of evolution equations that depend on an arbitrary function. Among such substitutions, there are infinitely many pairwise-nonequivalent ones.