  The system of two nonlinear coupled oscillators is studied. As partial case
this system of equation is reduced to the Duffing oscillator which has many
applications for describing physical processes. It is well known that the
inverse scattering transform is one of the most powerful methods for solving
the Cauchy problems of partial differential equations. To solve the Cauchy
problem for nonlinear differential equations we can use the Lax pair
corresponding to this equation. The Lax pair for ordinary differential or
systems or for system ordinary differential equations allows us to find the
first integrals, which also allow us to solve the question of integrability for
differential equations. In this report we present the Lax pair for the system
of coupled oscillators. Using the Lax pair we get two first integrals for the
system of equations. The considered system of equations can be also reduced to
the fourth-order ordinary differential equation and the Lax pair can be used
for the ordinary differential equation of fourth order. Some special cases of
the system of equations are considered.
