Vestnik of Samara University. Natural Science SeriesVestnik of Samara University. Natural Science Series2541-75252712-8954Samara National Research University479210.18287/2541-7525-2012-18-9-136-150NEW CASE OF INTEGRABILITY IN DYNAMICS OF MULTI-DIMENSIONAL BODYPokhodnyaN.V.morenov.sv@ssau.ruShamolinM.V.morenov.sv@ssau.ruMoscow Pedagogical State UniversityMoscow State University101220121891361500806201708062017Copyright © 2012, Pokhodnya N., Shamolin M.2012In this chapter the new results are systematized on study of the equations of motion of dynamically symmetrical four-dimensional (4D—) rigid body which residing in a certain nonconservative field of forces in case of special dynamical symmetry. Its type is unoriginal from dynamics of the real smaller-dimensional rigid bodies of interacting with a resisting medium on the laws of a jet flow, under which the nonconservative tracing force acts onto the body and forces both the value of velocity of a certain typical point of the rigid body and the certain phase variable to remain as constant in all time, that means the presence in system nonintegrable servo-constraints.многомерное твердое тело, интегрируемость, трансцендентный первый интеграл.multi-dimensional rigid body, integrability, transcendental first integral