  In this paper we study spectral properties of a family of quasi-periodic
Schr\"odinger operators on the real line in the adiabatic limit. We assume that
the adiabatic iso-energetic curve has a real branch that is extended along the
momentum direction. In the energy intervals where this happens, we obtain an
asymptotic formula for the Lyapunov exponent and show that the spectrum is
purely singular. This result was conjectured and proved in a particular case by
Fedotov and Klopp in \cite{FEKL1}.
