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 ≤ 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 u t = F (x, t, u, ∂u/∂x,..., ∂n u/∂ x n) that are conditionally invariant under a given generalized (Lie-Bäcklund) vector field Q(x, t, u, ∂u/∂x,..., ∂k u/∂x k)∂/∂u under the assumption that the system of ODEs 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.