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We propose prompt searches for X-ray and optical counterparts to IceCube high-energy astrophysical neutrinos with INTEGRAL. The astrophysical flux of high-energy neutrinos discovered by IceCube (Aarsten et al. 2013 Science 22 342) is expected to help us reveal the elusive sources of cosmic rays. While no neutrino point-sources have been detected so far a significant increase in the sensitivity of this study can be obtained by performing electromagnetic (EM) observations of IceCube neutrino positions to search for transient emitters. The IceCube observatory has implemented a system of realtime alerts that will broadcast the positions of candidate astrophysical neutrinos to the community. We will use these public alerts to trigger target-of-opportunity (ToO) observations with the INTEGRAL satellite. [truncated! Please see actual data for full text]