  Semi-regular (SR) variables are not a homogeneous class and their variability
is often explained due to pulsations and/or binarity. This study focuses on
IRAS 19135+3937, an SRd variable with an infra-red excess indicative of a dusty
disc. A time-series of high-resolution spectra, UBV photometry as well as a
very accurate light curve obtained by the Kepler satellite, allowed us to study
the object in unprecedented detail. We discovered it to be a binary with a
period of 127 days. The primary has a low surface gravity and an atmosphere
depleted in refractory elements. This combination of properties unambiguously
places IRAS 19135+3937 in the subclass of post-Asymptotic Giant Branch stars
with dusty discs.
  We show that the light variations in this object can not be due to
pulsations, but are likely caused by the obscuration of the primary by the
circumbinary disc during orbital motion. Furthermore, we argue that the
double-peaked Fe emission lines provide evidence for the existence of a gaseous
circumbinary Keplerian disc inside the dusty disc. A secondary set of
absorption lines has been detected near light minimum, which we attribute to
the reflected spectrum of the primary on the disc wall, which segregates due to
the different Doppler shift. This corroborates the recent finding that
reflection in the optical by this type of discs is very efficient. The system
also shows a variable Halpha profile indicating a collimated outflow
originating around the companion. IRAS 19135+3937 thus encompasses all the
major emergent trends about evolved disc systems, that will eventually help to
place these objects in the evolutionary context.
