Алгебраические и геометрические свойства квадратичных гамильтонианов, задаваемых секционными операторами

Following the terminology introduced by V. V. Trofimov and A. T. Fomenko, we say that a self-adjoint operator $\phi\colon \mathfrak{g}^* \to \mathfrak{g}$ is sectional if it satisfies the identity $\operatorname{ad}^{*}_{\phi x}a=\operatorname{ad}^{*}_{\beta}x$, $x\in \mathfrak{g}^*$, where $\mathfrak{g}$ is a finite-dimensional Lie algebra and $a\in \mathfrak{g}^*$ and $\beta \in \mathfrak{g}$ are fixed elements. In the case of a semisimple Lie algebra $\mathfrak{g}$, the above identity takes the form $[\phi x,a]=[\beta,x]$ and naturally arises in the theory of integrable systems and differential geometry (namely, in the dynamics of $n$-dimensional rigid bodies, the argument shift method, and the classification of projectively equivalent Riemannian metrics). This paper studies general properties of sectional operators, in particular, integrability and the bi-Hamiltonian property for the corresponding Euler equation $\dot x=\operatorname{ad}^*_{\phi x} x$.