  Existence of spherically symmetric solutions to the Einstein-Vlasov system is
well-known. However, it is an open problem whether or not static solutions
arise as minimizers of a variational problem. Apart from being of interest in
its own right, it is the connection to non-linear stability that gives this
topic its importance. This problem was considered in \cite{Wol}, but as has
been pointed out in \cite{AK}, the paper \cite{Wol} contained serious flaws. In
this work we construct static solutions by solving the Euler-Lagrange equation
for the energy density $\rho$ as a fixed point problem. The Euler-Lagrange
equation originates from the particle number-Casimir functional introduced in
\cite{Wol}. We then define a density function $f$ on phase space which induces
the energy density $\rho$ and we show that it constitutes a static solution of
the Einstein-Vlasov system. Hence we settle rigorously parts of what the author
of \cite{Wol} attempted to prove.
