The article studies the problem of the eXactfeedback linearization of the nonlinear control systems. This is a problem how to use the feedback controls in order to modify the original internal dynamics of a controlled system in such a way as to obtain to same behavior of certain prescribed autonomous linear systems. It presents possibilities of a successful selection of the linearizing feedback from the structure form of the initial nonlinear control system, as well as from the considered domain of the phase state. In this article the authors present a specific feature of construction of a linearizing feedback, when the initial nonlinear control system has the first integral and the control process takes start from the initial state where this first integral equals to zero. As an eXample a problem of three-aXis reorientation of a rigid spacecraft was considered. Three reaction wheels were employed to produce the necessary torque in the aXes of the initial. The controlling moments, applied to the flywheels, were offered to be generated by means of a feedback in the form of nonlinear functions of the phase variables of the considered nonlinear controlled system of differential equations, including dynamic Euler equations and kinematic equations in Rodrigues-Hamilton variables (in terms of the quaternion). As a result, the solution to the original nonlinear problem was narrowed down to the elementary linear control problems. The above-mentioned peculiarity in construction of the linearizing feedback takes place in this problem, when the initial angular velocities of the spacecraft and reaction wheels equal to zero.