10 namical system reads we expect chaotic energy exchange among the modes of such a cluster. The structure of resonance clustering is substantially richer in this case. For instance, 83 different vortici- ties can generate two-cluster clusters, among those most clusters have AP-connections but also clusters with AA- connections appear, for instance where the indices a , b and c are taken for three triads; all connections within this cluster are of AA-type, i.e. B$_{3}$$_{|}$$_{a}$ = B$_{3}$$_{|}$$_{b}$ = B$_{3}$$_{|}$$_{c}$ . The cluster of four triads has three AA-connections and one AP-connection. The NR-diagram (see Fig.9) and the dynamical system can be easily constructed but no results on the integrability of the corresponding dynamical system are known. It would be plausible to assume that in this situation no regular patterns will be observable for generic initial conditions. On the other hand, a special choice of initial conditions — excitation of a P-mode — might produce a fluxless (frozen) regime. However, one has to construct the resonance clustering and to check whether or not the "parasite" frequency of electronic equipment used during experiments generates a resonance (see [4] for more detail). Last but not least. Let us rewrite (37) as where does not depend on the properties of fluid: these are absorbed in the coefficient σ. The expression for ˜ ω ( k ) can also be rewritten as and others. The corresponding NR-diagram is shown in Fig.8. The explicit form of the dynamical system for a cluster con- sisting of a triad a and a triad b connected via two high- frequency modes, B$_{3}$$_{|}$$_{a}$ = B$_{3}$$_{|}$$_{b}$ , reads The integrability of (43) has been studied in [2, 49] with the following results. The system (43) is integrable for arbitrary initial conditions , if Z$_{a}$/Z$_{b}$ is equal to 1, 2 or 1/2, [49]. In the case of arbitrary Z$_{a}$/Z$_{b}$ , the integrability can be proven for some specific initial conditions [31]. Beside two-triad clusters we also observe the appear- ance of various structures formed by three, four, five and more connected triads; some examples are given below. The cluster of three triads has the NR-diagram shown in Fig.9; accordingly, its dy- with ˜ Z depending only on the wave numbers k$_{1}$, k$_{2}$, k$_{3}$. This means that the cluster integrability defined by the with ( | k | 3 + ˜ Ω $^{2}$/ 4) not depending on σ. Correspondingly, the interaction coefficient Z has the form Ω( k$_{1}$, k$_{2}$ ) = σ 1 / 2 · ˜ Ω( k$_{1}$, k$_{2}$ ) (47) ˜ Ω( k$_{1}$, k$_{2}$ ) = k$_{1}$k$_{2}$ (9 k 2 1 + 9 k 2 + 14 k$_{1}$k$_{2}$ )( k$_{1}$ + k$_{2}$ ) − 1 / 2 √ 6(6 k 4 1 + 15 k 3 $_{1}$k$_{2}$ + 22 k 2 $_{1}$k 2 + 15 k$_{1}$k 3 2 + 6 k 4 $_{2}$) ˜ ω ( k ) = √ σ | k | 3 + Ω 2 4 = σ 1 / 2 √$_{|}$ k | 3 + ˜ Ω 2 4 , (48) Z = σ 1 / 4 · ˜ Z (49) (79 , 80 , 159) , (78 , 81 , 159) , (77 , 82 , 159) (44)     ˙ B$_{1}$$_{|}$$_{a}$ = Z$_{a}$B ∗ 2 | $_{a}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{1}$$_{|}$$_{b}$ = Z$_{b}$B ∗ 2 | $_{b}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{2}$$_{|}$$_{a}$ = Z$_{a}$B ∗ 1 | $_{a}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{2}$$_{|}$$_{b}$ = Z$_{b}$B ∗ 1 | $_{b}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{3}$$_{|}$$_{a}$ = − Z$_{a}$B$_{1}$$_{|}$$_{a}$B$_{2}$$_{|}$$_{a}$ − Z$_{b}$B$_{1}$$_{|}$$_{b}$B$_{2}$$_{|}$$_{b}$. (43) FIG. 9: NR-diagram for the clusters given by (44) (on the left) and by (46) (on the right). FIG. 8: NR-diagram for clusters described by (42). FIG. 7: NR-diagram for clusters appearing for ε ≤ 10 − $^{4}$.      ˙ B$_{1}$$_{|}$$_{a}$ = Z$_{a}$B ∗ 2 | $_{a}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{2}$$_{|}$$_{a}$ = Z$_{a}$B ∗ 1 | $_{a}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{1}$$_{|}$$_{b}$ = Z$_{b}$B ∗ 2 | $_{b}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{2}$$_{|}$$_{b}$ = Z$_{b}$B ∗ 1 | $_{b}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{1}$$_{|}$$_{c}$ = Z$_{c}$B ∗ 2 | $_{c}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{2}$$_{|}$$_{c}$ = Z$_{c}$B ∗ 1 | $_{c}$B$_{3}$$_{|}$$_{a}$ , ˙ B$_{3}$$_{|}$$_{a}$ = − Z$_{a}$B$_{1}$$_{|}$$_{a}$B$_{2}$$_{|}$$_{a}$ − Z$_{b}$B$_{1}$$_{|}$$_{b}$B$_{2}$$_{|}$$_{b}$ − Z$_{c}$B$_{1}$$_{|}$$_{c}$B$_{2}$$_{|}$$_{c}$ (45) (48 , 48 , 96) , (47 , 49 , 96) , (28 , 96 , 124) , (46 , 50 , 96) (46)    $^{(50}$$^{,}$$^{50}$$^{,}$$^{100)}$$^{,}$$^{(49}$$^{,}$$^{51}$$^{,}$$^{100);}$ (47 , 48 , 95) , (46 , 49 , 95); (44 , 44 , 88) , (43 , 45 , 88) (42) • 10 − 4 ≤ ε ≤ 10 − $^{3}$. The size of the clusters grows exponentially with the growth of ε so that for ε = 10 − 2 some vorticities generate clusters formed by a few hundreds to few thousands of connected triads. The maximal cluster in the studied spectral domain consists of about 4000 triads with more than 33.000 connections among them. • 10 − 3 ≤ ε ≤ 10 − $^{2}$.