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During the first observing run of LIGO, two gravitational wave events and onelower-significance trigger (LVT151012) were reported by the LIGO/Virgocollaboration. At the time of LVT151012, the INTErnational Gamma-RayAstrophysics Laboratory (INTEGRAL) was pointing at a region of the skycoincident with the high localization probability area of the event and thuspermitted us to search for its electromagnetic counterpart (both prompt andafterglow emission). The imaging instruments on-board INTEGRAL (IBIS/ISGRI,IBIS/PICsIT, SPI, and the two JEM-X modules) have been exploited to attempt thedetection of any electromagnetic emission associated with LVT151012 over 3decades in energy (from 3 keV to 8 MeV). The omni-directional instrumentson-board the satellite, i.e. the SPI-ACS and IBIS monitored the entireLVT151012 localization region at energies above 75 keV. We did not find anysignificant transient source that was spatially and/or temporally coincidentwith LVT151012, obtaining tight upper limits on the associated hard X-ray and$\gamma$-ray radiation. For typical spectral models, the upper limits on thefluence of the emission from any 1 s long-lasting counterpart of LVT151012ranges 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), constrainingthe ratio of the isotropic equivalent energy released in the electromagneticemission to the total energy of the gravitational waves:$E_{75-2000~keV}/E_{GW}<$4.4$\times$10$^{-5}$. Finally, we provide anexhaustive summary of the capabilities of all instruments on-board INTEGRAL tohunt for $\gamma$-ray counterparts of gravitational wave events, exploitingboth serendipitous and pointed follow-up observations. This will serve as areference for all future searches.

INTEGRAL IBIS, SPI, and JEM-X observations of LVT151012