  In the history of mechanics, there have been two points of view for studying
mechanical systems: The Newtonian and the Cartesian.
  According the Descartes point of view, the motion of mechanical systems is
described by the first-order differential equations in the $N$ dimensional
configuration space $\textsc{Q}.$
  In this paper we develop the Cartesian approach for mechanical systems with
three degrees of freedom and with constraint which are linear with respect to
velocity. The obtained results we apply to discuss the integrability of the
geodesic flows on the surface in the three dimensional Euclidian space and to
analyze the integrability of a heavy rigid body in the Suslov and the Veselov
cases .
