Higher-order reductions of the Mikhalev system

We consider the 3D Mikhalev system, $$ u_t=w_x, \quad u_y= w_t-u w_x+w u_x, $$ which has first appeared in the context of KdV-type hierarchies. Under the reduction $w=f(u)$, one obtains a pair of commuting first-order equations, $$ u_t=f'u_x, \quad u_y=(f'^2-uf'+f)u_x, $$ which govern simple wave solutions of the Mikhalev system. In this paper we study {\it higher-order} reductions of the form $$ w=f(u)+εa(u)u_x+ε^2[b_1(u)u_{xx}+b_2(u)u_x^2]+..., $$ which turn the Mikhalev system into a pair of commuting higher-order equations. Here the terms at $ε^n$ are assumed to be differential polynomials of degree $n$ in the $x$-derivatives of $u$. We will view $w$ as an (infinite) formal series in the deformation parameter $ε$. It turns out that for such a reduction to be non-trivial, the function $f(u)$ must be quadratic, $f(u)=λu^2$, furthermore, the value of the parameter $λ$ (which has a natural interpretation as an eigenvalue of a certain second-order operator acting on an infinite jet space), is quantised. There are only two positive allowed eigenvalues, $λ=1$ and $λ=3/2$, as well as infinitely many negative rational eigenvalues. Two-component reductions of the Mikhalev system are also discussed. We emphasise that the existence of higher-order reductions of this kind is a reflection of {\it linear degeneracy} of the Mikhalev system, in particular, such reductions do not exist for most of the known 3D dispersionless integrable systems such as the dispersionless KP and Toda equations.