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This paper is concerned with the analysis of a mathematical model arising inplasma physics, more specifically in fusion research. It directly follows\cite{DHK1}, where the tri-dimensional analysis of a Vlasov-Poisson equationwith finite Larmor radius scaling was led, corresponding to the case of ionswith massless electrons whose density follows a linearized Maxwell-Boltzmannlaw. We now consider the case of electrons in a background of fixed ions, whichwas only sketched in \cite{DHK1}. Unfortunately, there is evidence that theformal limit is false in general. Nevertheless, we formally derive a fluidsystem for particular monokinetic data. We prove the local in time existence ofanalytic solutions and rigorously study the limit (when the Debye lengthvanishes) to a new anisotropic fluid system. This is achieved thanks toCauchy-Kovalevskaya type techniques, as introduced by Caflisch \cite{Caf} andGrenier \cite{Gre1}.

On the three-dimensional finite Larmor radius approximation: the case of electrons in a fixed background of ions