  We show how to extend (and with what limitations) Avila's global theory of
analytic SL(2,C) cocycles to families of cocycles with singularities. This
allows us to develop a strategy to determine the Lyapunov exponent for extended
Harper's model, for all values of parameters and all irrational frequencies. In
particular, this includes the self-dual regime for which even heuristic results
did not previously exist in physics literature. The extension of Avila's global
theory is also shown to imply continuous behavior of the LE on the space of
analytic $M_2(\mathbb{C})$-cocycles. This includes rational approximation of
the frequency, which so far has not been available.
