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A case of complete integrability in the dynamics on the tangent bundle of a two-dimensional sphere

2. The system of equations has variable dissipation with zero mean. The system of equations (1)–(3) is a dynamical system having variable dissipation with zero mean (in this case the mean is taken with respect to the angle α) [2]. This expresses the fact that the integral of the divergence of the right-hand side of the system over a period of the attack angle is zero (after an appropriate reduction of the system, this integral reflects the variation of the phase volume). So the system is, in a sense, ‘semiconservative’. If z1 = Ωy cos β1 + Ωz sin β1, z2 = −Ωy sin β1 + Ωz cos β1, σh1/I2 = H1, β = σAB/I2, and σzk = νwk, k = 1, 2 (here α ′ = να•/σ, and so on), then we obtain a supporting system of the form