  The Noether-like operators that play an essential role in writing down the
invariants for systems of two ordinary differential equations (ODEs) are
constructed. The classification of such operators is carried out with the help
of analytic continuation of the Lagrangians on the line. Cases in which the
Noether-like operators are also Noether symmetries for the systems of ODEs are
briefly mentioned. In particular, the 8-dimensional maximal Noether subalgebra
is remarkabely obtained for the simplest system of the free particle equations
in two dimensions from the 5-dimensional complex Noether algebra. We present
the effectivness of Noether-like operators as well as the determination of all
first integrals of systems of nonlinear differential equation which have not
been reported before. This study gives a new direction to construct first
integrals for systems of nonlinear differential equations.
