  Coupled second order nonlinear differential equations are of fundamental
importance in dynamics. In this part of our study on the integrability and
linearization of nonlinear ordinary differential equations we focus our
attention on the method of deriving general solution of two coupled second
order nonlinear ordinary differential equations through the extended
Prelle-Singer procedure. We describe a procedure to obtain integrating factors
and required number of integrals of motion so that the general solution follows
straightforwardly from these integrals. Our method tackles both isotropic and
non-isotropic cases in a systematic way. In addition to the above, we introduce
a new method of transforming coupled second order nonlinear ODEs into uncoupled
ones. We illustrate the theory with potentially important examples.
