  Consider a balanced non triangular two-color P\'olya-Eggenberger urn process,
assumed to be large which means that the ratio sigma of the replacement matrix
eigenvalues satisfies 1/2<sigma <1. The composition vector of both discrete
time and continuous time models admits a drift which is carried by the
principal direction of the replacement matrix. In the second principal
direction, this random vector admits also an almost sure asymptotics and a
real-valued limit random variable arises, named WDT in discrete time and WCT in
continous time. The paper deals with the distributions of both W. Appearing as
martingale limits, known to be nonnormal, these laws remain up to now rather
mysterious.
  Exploiting the underlying tree structure of the urn process, we show that WDT
and WCT are the unique solutions of two distributional systems in some suitable
spaces of integrable probability measures. These systems are natural extensions
of distributional equations that already appeared in famous algorithmical
problems like Quicksort analysis. Existence and unicity of the solutions of the
systems are obtained by means of contracting smoothing transforms. Via the
equation systems, we find upperbounds for the moments of WDT and WCT and we
show that the laws of WDT and WCT are moment-determined. We also prove that WDT
is supported by the whole real line and admits a continuous density (WCT was
already known to have a density, infinitely differentiable on R\{0} and not
bounded at the origin).
