Periodic orbits of the three dimensional logarithm galactic potential

We apply the averaging theory of first order to study analytically families of periodic orbits for a three dimensional logarithmic galactic potential H = 1/2(p(x)(2) + p(y)(2) + p(z)(2)) + v(0)(2)/2 ln(x(2 )- lambda x(3) + alpha y(2 )+ bz(2) + c(b)(2)), that is relevant in the study of elliptic galactic dynamics. We first introduce a scale transformation in the coordinates and momenta with a parameter epsilon and we find, using averaging theory of first order in epsilon, the existence up to three periodic orbits if alpha, beta are irrational, and one periodic orbit if either alpha is irrational and beta is rational, beta or is irrational and alpha is rational, for epsilon sufficiently small.