  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 $$ w=f(u)+\epsilon a(u)u_x+\epsilon^2[b_1(u)u_{xx}+b_2(u)u_x^2]+...,
$$ which turn the Mikhalev system into a pair of commuting higher-order
equations. Here the terms at $\epsilon^n$ 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 $\epsilon$. It turns out
that for such a reduction to be non-trivial, the function $f(u)$ must be
quadratic, $f(u)=\lambda u^2$, furthermore, the value of the parameter
$\lambda$ (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, $\lambda=1$ and $\lambda=3/2$, 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.
