  In this work we study the so-called ModMax nonlinear electrodynamics, which
is a novel model designed to preserve duality rotations and conformal
transformations, such as the Maxwell's equations do. This model allows to study
diverse gravitational phenomena when is coupled to General Relativity, in
particular charged black holes and gravitational waves. In the present work we
focus in the dynamics and Hamiltonian analysis of the model. Specifically, we
analyze the propagation of the discontinuities of the field and obtain the
corresponding dispersion relations. To perform the Hamiltonian analysis we
adopt the first order formalism develop by Pleba\'nski and follow the Dirac
method for theories with constraints. We derive the effective Hamiltonian,
classify all the constraints and identify the degrees of freedom. We prove that
the effective Hamiltonian is strictly bounded from below and investigate the
existence of non trivial minima.
