  We show that any non-degenerate vector field $u$ in $ L^{\infty}(\Omega,
\R^N)$, where $\Omega$ is a bounded domain in $\R^N$, can be written as
{equation} \hbox{$u(x)= \nabla_1 H(S(x), x)$ for a.e. $x \in \Omega$},
{equation} where $S$ is a measure preserving point transformation on $\Omega$
such that $S^2=I$ a.e (an involution), and $H: \R^N \times \R^N \to \R$ is a
globally Lipschitz anti-symmetric convex-concave Hamiltonian. Moreover, $u$ is
a monotone map if and only if $S$ can be taken to be the identity, which
suggests that our result is a self-dual version of Brenier's polar
decomposition for the vector field $u$ as $u(x)=\nabla \phi (S(x))$, where
$\phi$ is convex and $S$ is a measure preserving transformation. We also
describe how our polar decomposition can be reformulated as a self-dual mass
transport problem.
