  We study nonlinear vacuum electrodynamics in a first-order formulation
proposed by Pleba\'nski. By applying a Dirac constraint analysis, we derive an
effective Hamiltonian, together with the equations of motion. We show that
there exists a large class of potentials for which the effective Hamiltonian is
bounded from below, while at the same time possessing stationary points in
which the field strength acquires a nonzero vacuum expectation value. The
associated spontaneous breaking of Lorentz symmetry can in principle be
detected by coupling the model to a suitable external current, or to gravity.
We show that the possible vacua can be classified in four classes. We study
some of their properties, using explicit examples for illustration.
