  In this work we consider formal singular vector fields in $ C^{3}$with an
isolated and doubly-resonant singularity of saddle-node typeat the origin. Such
vector fields come from irregular two-dimensionalsystems with two opposite
non-zero eigenvalues, and appear for instancewhen studying the irregular
singularity at infinity in Painlev{\'e}
equations$(P\_{j})\_{j\in(I,II,III,IV,V)}$, for generic values of the
parameters.Under generic assumptions we give a complete formal
classificationfor the action of formal diffeomorphisms (by changes of
coordinates)fixing the origin and fibered in the independent variable. Wealso
identify all formal isotropies (self-conjugacies) of the normalforms. In the
particular case where the flow preserves a transversesymplectic structure, e.g.
for Painlev{\'e} equations, we provethat the normalizing map can be chosen to
preserve the transversesymplectic form.
