  We consider a singular Sturm-Liouville expression with the indefinite weight
sgn x. To this expression there is naturally a self-adjoint operator in some
Krein space associated. We characterize the local definitizability of this
operator in a neighbourhood of $\infty$. Moreover, in this situation, the point
$\infty$ is a regular critical point. We construct an operator $A=(\sgn
x)(-d^2/dx^2+q)$ with non-real spectrum accumulating to a real point. The
obtained results are applied to several classes of Sturm-Liouville operators.
