Geodesic
Differential substitutions of the Miura transformation type
Teoretičeskaâ i matematičeskaâ fizika, Tome 116 (1998) no. 3, pp. 336-348

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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. 
@article{TMF_1998_116_3_a1,
     author = {S. Ya. Startsev},
     title = {Differential substitutions of the {Miura} transformation type},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {336--348},
     publisher = {mathdoc},
     volume = {116},
     number = {3},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1998_116_3_a1/}
}
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S. Ya. Startsev. Differential substitutions of the Miura transformation type. Teoretičeskaâ i matematičeskaâ fizika, Tome 116 (1998) no. 3, pp. 336-348. http://geodesic.mathdoc.fr/item/TMF_1998_116_3_a1/