A note on the invariants of the Volterra gyrostat and each of its known special cases, and implications for low order models∗

The paper derives the two invariants for the energy-conserving tVG (C1 = 1/2 Σ x_i^2 and C2 in Eq. (38)), relates C2 to M − K1^2E (Eq. (40)), and characterizes invariants for all subclasses via a linear system (Eq. (29)), including the non–energy-conserving case where the general model has no nontrivial quadratic–affine invariant; subclasses with two or one linear terms have one or two invariants respectively . The candidate solution reproduces these results by the same null-space conditions and explicit coefficient solving, with a small but immaterial additive-constant correction in the identification of C2 with M − K1^2E (a 1/2 factor on Σ h_i^2) consistent with invariance up to constants .