  We study the normal forms for incompressible flows and maps in the
neighborhood of an equilibrium or fixed point with a triple eigenvalue. We
prove that when a divergence free vector field in $\mathbb{R}^3$ has nilpotent
linearization with maximal Jordan block then, to arbitrary degree, coordinates
can be chosen so that the nonlinear terms occur as a single function of two
variables in the third component. The analogue for volume-preserving
diffeomorphisms gives an optimal normal form in which the truncation of the
normal form at any degree gives an exactly volume-preserving map whose inverse
is also polynomial inverse with the same degree.
