= Solution
Take $k>0$ initially; real solutions for $k<0$ follow by $(k,\omega)\mapsto(-k,-\omega)$. Put $V=v_0+v_2-v_1$. Approaching the real <phase velocity> from the upper half-plane, the <Landau contour> and the <Sokhotski–Plemelj formula> give the following <Cauchy principal value> decomposition
$$
\int_{C_L}\frac{f_0'(v)}{v-u}\,dv=\operatorname{PV}\int\frac{f_0'(v)}{v-u}\,dv+i\pi f_0'(u).
$$
The real <frequency> condition $\epsilon=0$ therefore requires $f_0'(u)=0$. This is possible in the interior of the <velocity plateau> and outside the occupied <velocity> range:
$$
\boxed{u<-v_0,\qquad v_1<u<v_2,\qquad u>V.}
$$
These are candidate intervals, not yet solutions of the real <dispersion relation>. The endpoints have jumps in $f_0'$ and logarithmic singularities in the response; they are not finite-$k$ undamped modes. At the cusp $u=0$, the slope is also discontinuous and the real response diverges, so it supplies no additional mode.
The physical condition is absence of net resonant energy exchange with particles moving at the wave's <phase velocity>. Outside the support there are no such particles. Within a <velocity plateau>, the equal populations on either side of the resonance cancel <Landau damping>.
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