  The cubic Szego equation has been studied as an integrable model for
deterministic turbulence, starting with the foundational work of Gerard and
Grellier. We introduce a truncated version of this equation, wherein a majority
of the Fourier mode couplings are eliminated while the signature features of
the model are preserved, namely, a Lax-pair structure and a nested hierarchy of
finite-dimensional dynamically invariant manifolds. Despite the impoverished
structure of the interactions, the turbulent behaviors of our new equation are
stronger in an appropriate sense than for the original cubic Szego equation. We
construct explicit analytic solutions displaying exponential growth of Sobolev
norms. We furthermore introduce a family of models that interpolate between our
truncated system and the original cubic Szego equation, along with a few other
related deformations. All of these models possess Lax pairs, invariant
manifolds, and display a variety of turbulent cascades. We additionally mention
numerical evidence that shows an even stronger type of turbulence in the form
of a finite-time blow-up in some different, closely related dynamical systems.
