  We study the flow equations for BPS black holes in $\mathcal{N} = 2$
five-dimensional gauged supergravity coupled to any number of vector multiplets
via FI couplings. We develop the Noether-Wald procedure in this context and
exhibit the conserved charges as explicit integrals of motion, in the sense
that they can be computed at any radius on the rotating spacetime. The boundary
conditions needed to solve the first order differential equations are discussed
in great detail. We extremize the entropy function that controls the near
horizon geometry and give explicit formulae for all geometric variables at
their supersymmetric extrema. We have also considered a complexification of the
near-horizon variables that elucidates some features of the theory from the
near-horizon perspective.
