  We investigate the instability index of the spectral problem $$ -c^2y'' +
b^2y + V(x)y = -\mathrm{i} z y' $$
  on the line $\mathbb{R}$, where $V\in L^1_{\rm loc}(\mathbb{R})$ is real
valued and $b,c>0$ are constants. This problem arises in the study of stability
of solitons for certain nonlinear equations (e.g., the short pulse equation and
the generalized Bullough-Dodd equation). We show how to apply the standard
approach in the situation under consideration and as a result we provide a
formula for the instability index in terms of certain spectral characteristics
of the 1-D Schr\"odinger operator $H_V=-c^2\frac{d^2}{dx^2}+b^2 +V(x)$.
