  The extreme events are investigated for an $n$-component nonlinear
Schr\"odinger ($n$-NLS) system in the focusing Kerr-like nonlinear media, which
appears in many physical fields. We report and discuss the novel
multi-parametric families of vector rational rogue wave (RW) solutions
featuring the parity-time (PT) symmetry, which are characterized by
non-identical boundary conditions for the components, and consistent with the
degeneracy of $n$ branches of Benjamin-Feir instability. Explicit examples of
PT-symmetric vector RWs are presented. Some parameter constraints can make some
components generate the RWs with high amplitudes due to many-body resonant
interactions.Effect of a non-integrable deformation of the model on the
excitation of vector RWs is also discussed. These results will be useful to
design the RW experiments in multi-component physical systems.
