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Suppose that a rigid 4-dimensional body Θ with mass m and smooth 3-dimensional boundary ∂Θ has inertia tensor of the form diag{I1, I2, I2, I2} in some coordinate system Dx1x2x3x4 attached to Θ. The distance from the point D to the point N of application of a force S is a function depending on at least a certain angle α: DN = R(α, . . .) (cf. [1]–[3]). The force S has magnitude S = s(α) sgn cos α · v, |vD| = v, where s is a function characterizing both the energy dissipation and the energy pumping in the system [1], [2]. Here we take S = Sv(α) = Bv cos α, where B > 0. If Ω ∈ so(4) is the angular velocity tensor of Θ, then the part of the equations of motion corresponding to the Lie algebra so(4) has the following form [2]

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