  This article describes a purely analytic approach to urn models of the
generalized or extended P\'olya-Eggenberger type, in the case of two types of
balls and constant ``balance,'' that is, constant row sum. The treatment starts
from a quasilinear first-order partial differential equation associated with a
combinatorial renormalization of the model and bases itself on elementary
conformal mapping arguments coupled with singularity analysis techniques.
Probabilistic consequences in the case of ``subtractive'' urns are new
representations for the probability distribution of the urn's composition at
any time n, structural information on the shape of moments of all orders,
estimates of the speed of convergence to the Gaussian limit and an explicit
determination of the associated large deviation function. In the general case,
analytic solutions involve Abelian integrals over the Fermat curve x^h+y^h=1.
Several urn models, including a classical one associated with balanced trees
(2-3 trees and fringe-balanced search trees) and related to a previous study of
Panholzer and Prodinger, as well as all urns of balance 1 or 2 and a sporadic
urn of balance 3, are shown to admit of explicit representations in terms of
Weierstra\ss elliptic functions: these elliptic models appear precisely to
correspond to regular tessellations of the Euclidean plane.
