We will consider the $N=2$ abelian gauged supergravities in $d=4, 5$ coupled to arbitrary vector- and hypermultiplets. By choosing a static and spherical or hyperbolic ansatz for the extremal black hole in $d=4$ and for the extremal black string in $d=5$, it is possible to derive a one-dimensional effective action whose variation yields all the equations of motion. In particular, one can express the scalar potential in terms of a superpotential and write the action as a sum of squares. This procedure leads to the BPS first-order flow equations whose solution is driven by the Hamilton-Jacobi principal function of the one-dimensional dynamical system defined by the effective lagrangian. The latter squaring of the action is not unique and, by rotating the fluxes in an appropriate way, it is also possible to derive a non-BPS flow.

Furthermore, we will consider the near-horizon configurations and the so-called attractor equations, namely the near-horizon limit of the first-order flow equations. In particular, for the black string we will present the general solution for the scalars and for the radii of the horizon geometry in terms of the fluxes, the gauging parameters and the CY intersections numbers. From this result it is possible to derive the central charge of the two-dimensional CFT that describes the black string in the infrared.

Joint work with Dietmar Klemm and Marco Rabbiosi, arXiv:1602.01334, arXiv:1610.07367.