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Since early 2016 the IceCube collaboration is promptly distributing alerts of high-energy neutrino detections in order to allow follow-up searches for possible counterparts at the neutrino position. Finding such counterparts would provide important clues on the origin of the high-energy neutrinos and by proxy on the nature of the high-energy cosmic ray accelerators one of the most important and long-standing puzzles of high-energy astrophysics. We ask for pointed ToO observation of the HESE (High Energy Starting Events) and EHE (Extremely High Energy) neutrinos detected by IceCube and reported either in pub-lic or in private GCNs. We will employ the best available knowledge of the INTEGRAL instruments and our expertise in multimessenger transient searches to hunt for a possible hard X-ray and gamma-ray counterparts. In absence of a detection we will set the most stringent upper limits.We intend to promptly communicate the results of our ToOs to the general community thus supporting additional multiwavelength follow-ups and increasing the visibility of the INTEGRAL contribution to the searches of multimessenger transients. [truncated! Please see actual data for full text]