We investigate the geodesic curves of the homogeneous Gödel-type space-times, which constitute a two-parameter (l and Ω) class of solutions presented to several theories of gravitation (general relativity, Einstein-Cartan and higher derivative). To this end, we first examine the qualitative properties of those curves by means of the introduction of an effective potential and then accomplish the analytical integration of the equations of motion. We show that some of the qualitative features of the free motion in GSdel's universe (l 2=2Ω2) are preserved in all space-times, namely: (a) the projections of the geodesics onto the 2-surface (r, φ) are simple closed curves (with some exceptions forl 2≥4Ω2), and (b) the geodesies for which the ratio of azimuthal angular momentum to total energy,γ, is equal to zero always cross the originr=0. However, two new cases appear: (i) radially unbounded geodesies withγ assuming any (real) value, which may occur only for the causal space-times (l 2≥4Ω2), and (ii) geodesies withγ bounded both below and above, which always occur for the circular family (l 2<0) of space-times.
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