  We propose a geometric method to study the residual symmetries in $N=2$,
$d=4$ $\text{U}(1)$ Fayet-Iliopoulos (FI) gauged supergravity. It essentially
involves the stabilization of the symplectic vector of gauge couplings (FI
parameters) under the action of the U-duality symmetry of the ungauged theory.
In particular we are interested in those transformations that act non-trivially
on the solutions and produce scalar hair and dyonic black holes from a given
seed. We illustrate the procedure for finding this group in general and then
show how it works in some specific models. For the prepotential $F=-iX^0X^1$,
we use our method to add one more parameter to the rotating Chow-Comp\`ere
solution, representing scalar hair.
