  The existence of a formal particular solution (family of solutions) of
oscillating type under certain conditions has been proved for the quasi-linear
ordinary differential equations system. The asymptotic nature of this solution
(the family of solutions) is investigated in two individual cases when all the
eigenvalues of the matrix of the linear homogeneous part of the shorten system
of differential equations 1) are not pure imaginary, 2) are simple and under
some additional assumptions.
