  We consider two exact solutions of Einstein's field equations corresponding
to a cylinder of dust with net zero angular momentum. In one of the cases, the
dust distribution is homogeneous, whereas in the other, the angular velocity of
dust particles is constant [1]. For both solutions we studied the junction
conditions to the exterior static vacuum Levi-Civita spacetime. From this study
we find an upper limit for the energy density per unit length $\sigma$ of the
source equal ${1\over 2}$ for the first case and ${1\over 4}$ for the second
one. Thus the homogeneous cluster provides another example [2] where the range
of $\sigma$ is extended beyond the limit value ${1\over 4}$ previously found in
the literature [3,4]. Using the Cartan Scalars technics we show that the
Levi-Civita spacetime gets an extra symmetry for $\sigma={1\over 2}$ or
${1\over 4}$. We also find that the cluster of homogeneous dust has a superior
limit for its radius, depending on the constant volumetric energy density
$\rho_0$.
