The aim of this work is to prove analytically the existence of symmetric
periodic solutions of the family of Hamiltonian systems with Hamiltonian
1/2(q_1^2+p_1^2)+1/2(q_2^2+p_2^2)+ a q_1^4+b
q_1^2q_2^2+c \q_2^4 with three real
parameters a, b and c. Moreover, we characterize the stability of these
periodic solutions as function of the parameters. Also, we find a
first-order analytical approach of these symmetric periodic solutions.
We emphasize that these families of periodic solutions are different
from those that exist in the literature.