  Abell 1201 is a massive galaxy cluster at z=0.169 with a brightest cluster
galaxy (BCG) that acts as a gravitational lens to a background source at
z=0.451. The lensing configuration is unusual, with a single bright arc formed
at small radius $\sim$2 arcsec), where stars and dark matter are both expected
to contribute substantially to the total lensing mass. Here, we present deep
spectroscopic observations of the Abell 1201 BCG with MUSE, which reveal
emission lines from a faint counter-image, opposite to the main arc, at a
radius of 0.6 arcsec. We explore models in which the lensing mass is described
by a combination of stellar mass and a standard dark-matter halo. The
counter-image is not predicted in such models, unless the dark-matter component
is negligible, which would imply an extremely heavy stellar initial mass
function (IMF) in this galaxy. We consider two modifications to the model which
can produce the observed configuration without resorting to extreme IMFs.
Imposing a radial gradient in the stellar mass-to-light ratio, $\Upsilon$, can
generate a counter-image close to the observed position if $\Upsilon$ increases
by $\gtrsim$60 per cent within the inner $\sim$1 arcsec (e.g. variation from a
Milky-Way-like to a Salpeter-like IMF). Alternatively, the counter-image can be
produced by introducing a central super-massive black hole. The required mass
is $M_{\rm BH}$ = (1.3$\pm$0.6)$\times$10$^{10}$ M$_\odot$, which is comparable
to the largest black holes known to date, several of which are also hosted by
BCGs. We comment on future observations which promise to distinguish between
these alternatives.
