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Abstract

In this paper we consider the system of Hamiltonian differential equations, which determines small oscillations of a dynamical system with $n$ parameters. We demonstrate that this system determines an affinor structure $J$ on the phase space $T{\bf R}^n$. If $J^2 = \omega I$, where $\omega = \pm 1, 0$, the phase space can be considered as the biplanar space of elliptic, hyperbolic or parabolic type. In the Euclidean case (${\bf R}^n = E^n$) we obtain the Hopf bundle and its analogs. The bases of these bundles are, respectively, the projective $(n-1)$-dimensional spaces over algebras of complex, double and dual numbers.