  An amended MaxEnt formulation for systems displaced from the conventional
MaxEnt equilibrium is proposed. This formulation involves the minimization of
the Kullback-Leibler divergence to a reference $Q$ (or maximization of Shannon
$Q$-entropy), subject to a constraint that implicates a second reference
distribution $P\_{1}$ and tunes the new equilibrium. In this setting, the
equilibrium distribution is the generalized escort distribution associated to
$P\_{1}$ and $Q$. The account of an additional constraint, an observable given
by a statistical mean, leads to the maximization of R\'{e}nyi/Tsallis
$Q$-entropy subject to that constraint. Two natural scenarii for this
observation constraint are considered, and the classical and generalized
constraint of nonextensive statistics are recovered. The solutions to the
maximization of R\'{e}nyi $Q$-entropy subject to the two types of constraints
are derived. These optimum distributions, that are Levy-like distributions, are
self-referential. We then propose two `alternate' (but effectively computable)
dual functions, whose maximizations enable to identify the optimum parameters.
Finally, a duality between solutions and the underlying Legendre structure are
presented.
