ON ZERO-CURVATURE REPRESENTATIONS OF EVOLUTION-EQUATIONS

For zero-curvature representations (ZCRs) A(t)-B-x-AB+BA=0 of evolution equations u(t) = f(x, u, u(x),..., u(x...x)), we develop a description which is invariant under gauge transformations A' = SAS(-1) - SxS-1 and B' = SBS-1 - StS-1, where A, B and S are matrix functions of x, u, u(x), u(xx),.... We prove that every fixed matrix A of any dimension and order (in u(x...x)) determines a continual class of evolution equations which admit ZCRS with this A. Then we quote examples illustrating how a dependence of A on an essential parameter restricts classes of represented equations. One of our examples shows that some nonintegrable systems can admit parametric Lax pairs and infinitely many non-trivial conservation laws.