  During the first observing run of LIGO, two gravitational wave events and one
lower-significance trigger (LVT151012) were reported by the LIGO/Virgo
collaboration. At the time of LVT151012, the INTErnational Gamma-Ray
Astrophysics Laboratory (INTEGRAL) was pointing at a region of the sky
coincident with the high localization probability area of the event and thus
permitted us to search for its electromagnetic counterpart (both prompt and
afterglow emission). The imaging instruments on-board INTEGRAL (IBIS/ISGRI,
IBIS/PICsIT, SPI, and the two JEM-X modules) have been exploited to attempt the
detection of any electromagnetic emission associated with LVT151012 over 3
decades in energy (from 3 keV to 8 MeV). The omni-directional instruments
on-board the satellite, i.e. the SPI-ACS and IBIS monitored the entire
LVT151012 localization region at energies above 75 keV. We did not find any
significant transient source that was spatially and/or temporally coincident
with LVT151012, obtaining tight upper limits on the associated hard X-ray and
$\gamma$-ray radiation. For typical spectral models, the upper limits on the
fluence of the emission from any 1 s long-lasting counterpart of LVT151012
ranges from $F_{\gamma}=$3.5$\times$10$^{-8}$ erg cm$^{-2}$ (20 - 200 keV) to
$F_{\gamma}$=7.1$\times$10$^{-7}$ erg cm$^{-2}$ (75 - 2000 keV), constraining
the ratio of the isotropic equivalent energy released in the electromagnetic
emission to the total energy of the gravitational waves:
$E_{75-2000~keV}/E_{GW}<$4.4$\times$10$^{-5}$. Finally, we provide an
exhaustive summary of the capabilities of all instruments on-board INTEGRAL to
hunt for $\gamma$-ray counterparts of gravitational wave events, exploiting
both serendipitous and pointed follow-up observations. This will serve as a
reference for all future searches.
