Abstract
We study the Hess--Appelrot case of the Euler--Poisson system whichdescribes dynamics of a rigid body about a fixed point. We prove existenceof an invariant torus which supports hyperbolic or parabolic or ellipticperiodic or elliptic quasi--periodic dynamics. In the elliptic cases westudy the question of normal hyperbolicity of the invariant torus in thecase when the torus is close to a `critical circle'. It turns out that thenormal hyperbolicity takes place only in the case of $1:q$ resonance. In thesequent paper [16] we study limit cycles which appear afterperturbation of the above situation.
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