In this paper we discuss about the possibility of { \it coexistence} of stable and unstable quasi--periodic { \sc kam} tori in a region of phase space of the three-body problem. The { argument of proof} goes along { { \sc kam} theory and, especially,} the production of two non smoothly related systems of canonical coordinates in the same region of the phase space, the possibility of which is foreseen, for `properly--degenerate' systems, by a theorem of Nekhorossev and Mi{ \v{ s}}{ \v{ c}}enko and Fomenko. The two coordinate systems are alternative to the classical reduction of the nodes by Jacobi, described, e.g., in~[V.I.~Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, 18, 85 (1963); p. 141].
Categories: math.DS