Static cylindrically symmetric spacetimes

We prove the existence of static solutions to the cylindrically symmetric Einstein-Vlasov system, and we show that the matter cylinder has finite extension in two of the three spatial dimensions. The same results are also proved for a quite general class of equations of state for perfect fluids coupled to the Einstein equations, extending the class of equations of state considered by Bicak et al (2004 Class. Quantum Grav. 21 1583). We also obtain this result for the Vlasov-Poisson system