  Continuing our study on the complete integrability of nonlinear ordinary
differential equations, in this paper we consider the integrability of a system
of coupled first order nonlinear ordinary differential equations (ODEs) of both
autonomous and non-autonomous types. For this purpose, we modify the original
Prelle-Singer procedure so as to apply it to both autonomous and non-autonomous
systems of coupled first order ODEs. We briefly explain the method of finding
integrals of motion (time independent as well as time dependent integrals) for
two and three coupled first order ODEs by extending the Prelle-Singer(PS)
method. From this we try to answer some of the open questions in the original
PS method. We also identify integrable cases for the two dimensional
Lotka-Volterra system and three-dimensional R$\ddot{o}$ssler system as well as
other examples including non-autonomous systems in a straightforward way using
this procedure. Finally, we develop a linearization procedure for coupled first
order ODEs.
