We investigate the geodesics kinematics and dynamics in the Linet-Tian metric with Lambda<0 and compare with the results for the Levi-Civita metric, when Lambda=0. This is used to derive new stability results about the geodesics dynamics in static vacuum cylindrically symmetric spacetimes with respect to the introduction of Lambda<0. In particular, we find that increasing |Lambda| always increases the minimum and maximum radial distances to the axis of any spatially confined planar null geodesic. Furthermore, we show that, in some cases, the inclusion of any Lambda<0 breaks the geodesics orbit confinement of the Lambda=0 metric, for both planar and non-planar null geodesics, which are therefore unstable. Using the full system of geodesics equations, we provide numerical examples which illustrate our results.
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