On the classification of conditionally integrable evolution systems in $(1+1)$ dimensions

We generalize earlier results of Fokas and Liu and find all locally analytic $(1+1)$-dimensional evolution equations of order $n$ that admit an $N$-shock-type solution with $N\leq n+1$. For this, we develop a refinement of the technique from our earlier work, where we completely characterized all $(1+1)$-dimensional evolution systems $\boldsymbol{u}_t=\boldsymbol{F}(x,t,\boldsymbol{u},\partial\boldsymbol{u}/\partial x,\ldots,\partial^n\boldsymbol{u}/\partial x^n)$ that are conditionally invariant under a given generalized (Lie–Bäcklund) vector field $\boldsymbol{Q}(x,t,\boldsymbol{u},\partial\boldsymbol{u}/\partial x,\ldots,\partial^k\boldsymbol{u}/\partial x^k)\partial/\partial\boldsymbol{u}$ under the assumption that the system of ODEs $\boldsymbol{Q}=0$ is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in $t$, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.