  It is well known that the linear stability of solutions of partial
differential equations which are integrable can be very efficiently
investigated by means of spectral methods. We present here a direct
construction of the eigenmodes of the linearized equation by using only their
associated Lax pair with no reference to spectral data and boundary conditions.
This local construction is given in the general $N \times N$ matrix scheme so
as to be applicable to a large class of integrable equations, including the
multicomponent nonlinear Schroedinger system and the multi-wave resonant
interaction system. The analytical and numerical computations involved in this
general approach are detailed as an example for $N = 3$ for the particular
system of two coupled nonlinear Schroedinger equations in the defocusing,
focusing and mixed regimes. The instabilities of the continuous wave solutions
are fully discussed in the entire parameter space of their amplitudes and wave
numbers. By defining and computing the spectrum in the complex plane of the
spectral variable, the eigenfrequencies are explicitly expressed. According to
their topological properties, the complete classification of these spectra in
the parameter space is presented and graphically displayed. The continuous wave
solutions are linearly unstable for a generic choice of the coupling constants.
