  The authors have conjectured (\cite{KoM}) that if a normalized generalized
modular function (GMF) $f$, defined on a congruence subgroup $\Gamma$, has
integral Fourier coefficients, then $f$ is classical in the sense that some
power $f^m$ is a modular function on $\Gamma$. A strengthened form of this
conjecture was proved (loc cit) in case the divisor of $f$ is \emph{empty}. In
the present paper we study the canonical decomposition of a normalized
parabolic GMF $f = f_1f_0$ into a product of normalized parabolic GMFs $f_1,
f_0$ such that $f_1$ has \emph{unitary character} and $f_0$ has \emph{empty
divisor}. We show that the strengthened form of the conjecture holds if the
first "few" Fourier coefficients of $f_1$ are algebraic. We deduce proofs of
several new cases of the conjecture, in particular if either $f_0=1$ or if the
divisor of $f$ is concentrated at the cusps of $\Gamma$.
