  In this paper, we study the existence of non-planar periodic solutions for
the following spatial restricted 3-body and 4-body problems: for $N=2 or 3$,
given any masses $m_{1},...,m_{N}$, the mass points of $m_{1},...,m_{N}$ move
on the $N$ circular obits centered at the center of masses, the sufficiently
small mass moves on the perpendicular axis passing the center of masses. Using
variational minimizing methods, we establish the existence of the minimizers of
the Lagrangian action on anti-T/2 or odd symmetric loop spaces. Moreover, we
prove these minimizers are non-planar periodic solutions by using the Jacobi's
Necessary Condition for local minimizers.
