  The BPS equations in M-theory for solutions with 16 residual supersymmetries,
$SO(2,2)\times SO(4)\times SO(4)$ symmetry, and $AdS_4 \times S^7$ asymptotics,
were reduced in [arXiv:0806.0605] to a linear first order partial differential
equation on a Riemann surface with boundary, subject to a non-trivial quadratic
constraint. In the present paper, suitable regularity and boundary conditions
are imposed for the existence of global solutions. We seek regular solutions
with multiple distinct asymptotic $AdS_4 \times S^7$ regions, but find that,
remarkably, such solutions invariably reduce to multiple covers of the M-Janus
solution found by the authors in [arXiv:0904.3313], suggesting rigidity of the
half-BPS M-Janus solution. In particular, we prove analytically that no other
smooth deformations away from the M-Janus solution exist, as such deformations
invariably violate the quadratic constraint. These rigidity results are
contrasted to the existence of half-BPS solutions with non-trivial 4-form
fluxes and charges asymptotic to $AdS_7 \times S^4$. The results are related to
the possibility of M2-branes to end on M5-branes, but the impossibility of
M5-branes to end on M2-branes, and to the non-existence of half-BPS solutions
with simultaneous $AdS_4 \times S^7$ and $AdS_7 \times S^4$ asymptotic regions.
