الفهرس 
Introduction
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Abstract
Since there is no test for a mechanical system to be integrable, it is of great importance to construct new integrable systems and to tabulate all ones of certain types. In this thesis, we dealt with two different problems: Problem1: The method of construction of integrable system with a second integral polynomial in velocities on a Riemannian two dimensional manifold, developed by Yehia is used in this thesis to construct system of Kovalevskaya type. This problem is characterized by a potential of the structure and Applied to the problem of the motion of a rigid body about a fixed point, this method has led to the construction of two new conditional integrable cases. These results are included in ”New conditional integrable cases of motion of a rigid body with Kovalevskaya’s configuration”, ” Journal of Physics A: Mathematical and Theoretical” Vol. 44, PP. 1-8, (2011). A more detailed paper is being prepared for publication elsewhere. Problem2: In the control theory, certain terms are added to equations of the motion in order to stabilize certain unstable motion. These terms are called control terms and the new system is called controlled system. In this problem, we will add six linear controls to Euler equation in Euler case and study the problem from another point of view. We will prove that the last system is likely integrable by using Painleve approach. We also introduce a transformation that connects between the controlled system and a gyrostat system. By using this transformation, integrals of motions are constructed. The new system can be considered a generalization of classical Euler case. We also show the solution is reduced to an elliptic function. These results are being prepared for publication.