A reversible system with a small parameter mu is considered. When mu =
0 the generating system has a periodic motion, symmetric to a fixed s
et of the system automorphism. It is shown that this periodic motion i
s continued with respect to a small parameter in the Poincare-unisolat
ed case when certain conditions are satisfied only on the generating s
ystem. Symmetric periodic solutions are constructed both for a non-res
onant and for a resonant system. In the plane unrestricted three-body
problem the small parameter is chosen to be the quantity characterizin
g the interaction between two bodies chosen from the three. It is show
n that in this problem there are solutions in which the body moves alo
ng curves close to circles. The problem of the applicability of the re
sult to a sun-earth-moon type is discussed.