  Two fundamental facts of the modern wave turbulence theory are 1) existence
of power energy spectra in $k$-space, and 2) existence of "gaps" in this
spectra corresponding to the resonance clustering. Accordingly, three wave
turbulent regimes are singled out: \emph{kinetic}, described by wave kinetic
equations and power energy spectra; \emph{discrete}, characterized by resonance
clustering; and \emph{mesoscopic}, where both types of wave field time
evolution coexist. In this paper we study integrable dynamics of resonance
clusters appearing in discrete and mesoscopic wave turbulent regimes. Using a
novel method based on the notion of dynamical invariant we establish that some
of the frequently met clusters are integrable in quadratures for arbitrary
initial conditions and some others -- only for particular initial conditions.
We also identify chaotic behaviour in some cases. Physical implications of the
results obtained are discussed.
