  The European Space Agency is studying two large-class missions bound to
operate in the decade of the 30s, and aiming at investigating the most
energetic and violent phenomena in the Universe. $Athena$ is poised to study
the physical conditions of baryons locked in large-scale structures from the
epoch of their formation, as well as to yield an accurate census of accreting
super-massive black holes down to the epoch of reionization; LISA will extend
the hunt for Gravitational Wave (GW) events to the hitherto unexplored mHz
regime. We discuss in this paper the science that their concurrent operation
could yield, and present possible $Athena$ observational strategies. We focus
on Super-Massive (M$\lesssim10^7\rm M_{\odot}$) Black Hole Mergers (SMBHMs),
potentially accessible to $Athena$ up to $z\sim2$. The simultaneous measurement
of their electro-magnetic (EM) and GW signals may enable unique experiments in
the domains of astrophysics, fundamental physics, and cosmography, such as the
magneto-hydrodynamics of fluid flows in a rapidly variable space-time, the
formation of coronae and jets in Active Galactic Nuclei, and the measurement of
the speed of GW, among others. Key to achieve these breakthrough results will
be the LISA capability of locating a SMBHM event with an error box comparable
to, or better than the field-of-view of the $Athena$ Wide Field Imager
($\simeq0.4\,$deg$^2$) and $Athena$ capability to slew fast to detect the
source during the inspiral phase and the post-merger phase. Together, the two
observatories will open in principle the exciting possibility of truly
concurrent EM and GW studies of the SMBHMs
