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Cosmic rays enter the Earth’s atmosphere with energies up to 1020 eV, producing extended air showers, which contain up to billions of secondary particles that can be detected on the ground. Inside these showers, hadrons with large transverse momentum may be produced, which subsequently decay into muons, also carrying large transverse momentum. These muons separate from the shower core while traveling to the ground, producing lateral separations from the core up to several hundred meters. Together with the dense muon core bundle, they produce distinctive double-track signatures in the IceCube Neutrino Observatory. The corresponding lateral separation distributions are a measure of the underlying transverse momentum of hadrons, and therefore yield important information on hadron production during the air shower development. In this work, the lateral separation of atmospheric muons, between 135m and approximately 450 m, is measured using three years of IceCube data, taken between May 2012 and May 2015. A dedicated Monte Carlo for the simulation of laterally separated muons from air showers is presented. This is used to develop selection criteria in order to isolate the events of interest from IceCube data. In addition, a specific double-track reconstruction is introduced, which enables a precise measurement of the lateral separation and arrival direction of muons. Using existing energy estimation methods, which are optimized for the reconstruction of this class of events, the first primary energy dependent analysis of the lateral separation distribution of muons far from the shower core is performed. After applying all selection criteria, 80951 events are left with an effective lifetime of 960 days. The resulting lateral separation distributions are studied with emphasis on the angular arrival directions and the transverse momentum of muons, which is estimated based on Monte Carlo simulations. In addition, the sensitivity of laterally separated muons on the mass composition of cosmic ray nuclei and the seasonal variations of the resulting muon fluxes over three years are studied. Finally, the prospects for a search of double-track signatures, produced by exotic particles in cosmic ray air showers, which are predicted by theories beyond the standard model, are discussed. It is shown that, considering recent constraints on existing theoretical models, such as supersymmetry, the expected fluxes are below 10-3 events per year. Hence, it is concluded that the observation of exotic double-tracks from air showers, using existing neutrino telescopes, is not feasible.