We study stationary solutions in the differential kinetic equation, which was introduced in for description of a local dual cascade wave turbulence. We give a full classification of single-cascade states in which there is a finite flux of only one conserved quantity. Analysis of the steady-state spectrum is based on a phase-space analysis of orbits of the underlying dynamical system. The orbits of the dynamical system demonstrate the blow-up behaviour which corresponds to a sharp front where the spectrum vanishes at a finite wave number. The roles of the KZ and thermodynamic scaling as intermediate asymptotic, as well as of singular solutions, are discussed.
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