  We consider localized modes (discrete breathers) of the discrete nonlinear
Schr\"{o}dinger equation
$i\frac{d\psi_n}{dt}=\psi_{n+1}+\psi_{n-1}-2\psi_n+\sigma|\psi_n|^2\psi_n$,
$\sigma=\pm1$, $n\in \mathbb{Z}$. We study the diversity of the steady-state
solutions of the form $\psi_n(t)=e^{i\omega t}v_n$ and the intervals of the
frequency, $\omega$, of their existence. The base for the analysis is provided
by the anticontinuous limit ($\omega$ negative and large enough) where all the
solutions can be coded by the sequences of three symbols "-", "0" and "+".
Using dynamical systems approach we show that this coding is valid for
$\omega<\omega^*\approx -3.4533$ and the point $\omega^*$ is a point of
accumulation of saddle-node bifurcations. Also we study other bifurcations of
intrinsic localized modes which take place for $\omega>\omega^*$ and give the
complete table of them for the solutions with codes consisting of less than
four symbols.
