  The partial case of the planar $N+1$ body problem, $N\ge2$, of the type of
planetary system with satellites is studied. One of the bodies (the Sun) is
assumed to be much heavier than the other bodies ("planets" and "satellites"),
moreover the planets are much heavier than the satellites, and the "years" are
much longer than the "months". The existence of at least $2^{N-2}$ smooth
2-parameter families of symmetric periodic solutions in a rotating coordinate
system is proved, such that the distances between each planet and its
satellites are much shorter than the distances between the Sun and the planets.
The existence of "gaps" in these families of solutions is proved, corresponding
to $k:(k+1)$ resonances of angular frequencies of planets' revolution around
the Sun. Generating symmetric periodic solutions are described. Sufficient
conditions for some periodic solutions to be orbitally stable in linear
approximation are given. The results are extended to a class of Hamiltonian
systems with slow and fast variables close to the systems of semidirect product
type.
