  We analyze a general class of locally supersymmetric, CP and modular
invariant models of lepton masses depending on two complex moduli taking values
in the vicinity of a fixed point, where the theory enjoys a residual symmetry
under a finite group. Like in models that depend on a single modulus, we find
that all physical quantities exhibit a universal scaling with the distance from
the fixed point. There is no dependence on the level of the construction, the
weights of matter multiplets and their representations, with the only
restriction that electroweak lepton doublets transform as irreducible triplets
of the finite modular group. Also the form of the kinetic terms, which here are
assumed to be neither minimal nor flavor blind, is irrelevant to the outcome.
The result is remarkably simple and the whole class of examined theories gives
rise to five independent patterns of neutrino mass matrices. Only in one of
them, the predicted scaling agrees with the observed neutrino mass ratios and
lepton mixing angles, exactly as in single modulus theories living close to
$\tau=i$.
