the components $M_1,M_2,M_3$, then equations describe some (generalized) problem of the rigid body dynamics, in which $H^{(2)}$ stands for the kinetic energy, $H^{(0)}=H^{(0)}({{\boldsymbol \alpha}}, {{\boldsymbol \beta}})$ is the potential energy.

The vector ${{\boldsymbol \omega}}=\partial H / \partial {\mathbf{M}}$ is called the angular velocity. The term $H^{(1)}$ in generates forces which are usually called gyroscopic. The canonical unit bases in $so(3)\cong {\mathbb{R}}^3 = \{\mathbf{M}\}$, ${\mathbb{R}}^3=\{{{\boldsymbol \alpha}}\}$ and ${\mathbb{R}}^3=\{{{\boldsymbol \beta}}\}$ are identified with some orthonormal basis $O\mathbf{e}_1\mathbf{e}_2\mathbf{e}_3$ fixed in the rotating body (the moving frame with the origin $O$ at the fixed point of the body).

For a generic case we suppose that ${{\boldsymbol \alpha}}{\times}{{\boldsymbol \beta}}\ne 0$. The phase space of equations is a mensional submanifold ${\mathcal{P}}^6$ in $\mathbf{v}^*$ defined as a common level of the Casimir functions (geometric integrals) $${{\boldsymbol \alpha}}{\bdot} {{\boldsymbol \alpha}}= a^2, \quad {{\boldsymbol \beta}}{\bdot} {{\boldsymbol \beta}}= b^2, \quad {{\boldsymbol \alpha}}{\bdot} {{\boldsymbol \beta}}= c \qquad (a \geqslant b > 0, \; |c| < a b).$$ On ${\mathcal{P}}^6$ there exists a natural symplectic structure compatible with the restriction of the Lie–Poisson brackets.

If $H^{(0)}$ is a linear function in ${{\boldsymbol \alpha}}, {{\boldsymbol \beta}}$ and essentially depends on both vectors, corresponding potential field is called a double field. In this case 