DOI 10.1007/s10958-015-2220-0 Journal of Mathematical Sciences, Vol. 204, No. 6, February, 2015 CLASSIFICATION OF INTEGRABLE CASES IN THE DYNAMICS OF A FOUR-DIMENSIONAL RIGID BODY IN A NONCONSERVATIVE FIELD IN THE PRESENCE OF A TRACKING FORCE M. V. Shamolin UDC 517; 531.01 Abstract. This pap er is a survey of integrable cases in the dynamics of a four-dimensional rigid b ody under the action of a nonconservative force field. We review b oth new results and results obtained earlier. The problems examined are