the Vlasov-Poisson system with the parameter $\lambda$. The solution $\zeta_0$, corresponding to $\lambda=0$, is known and we have $\mathfrak F(\zeta_0,0)=0$. The main part of the work consists in showing that the implicit function theorem can be applied. Then it follows that to each $\lambda\in (-\delta,\delta)$ there exists $\zeta_\lambda \in \mathcal X$ such that $\mathfrak F(\zeta_\lambda, \lambda)=0$. This collection $\zeta_\lambda$ of functions consequently solves the Vlasov-Poisson system and this solution is axially symmetric but not spherically symmetric. It is in the nature of this method that the obtained rotating solutions have small overall angular momentum.

In [@akr11] a similar method with a different set up has been used to show the existence of axially but not spherically symmetric, static solutions of the Einstein-Vlasov system. In this context it was used that the Vlasov-Poisson system is the non-relativistic limit of the Einstein-Vlasov system, in the sense that a solution of the Einstein-Vlasov system converges to a solution of the Vlasov-Poisson system if the speed of light $c$ goes to infinity. So besides $\lambda$, the speed of light $c$ has been introduced to the system as a second parameter. Perturbing off a spherically symmetric, static solution of the Vlasov-Poisson system in those two 