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We consider a normal complete rational variety with a torus action of complexity one. In the main results, we determine the roots of the automorphism group and give an explicit description of the root system of its semisimple part. The results are applied to the study of almost homogeneous varieties. For example, we describe all almost homogeneous (possibly singular) del Pezzo K * -surfaces of Picard number one and all almost homogeneous (possibly singular) Fano threefolds of Picard number onedoi:10.17323/1609-4514-2014-14-3-429-471 fatcat:6gbilhyr65fdxdowd3746i2hfa