  The Painlev\'e transcendents discovered at the turn of the XX century by pure
mathematical reasoning, have later made their surprising appearance -- much in
the way of Wigner's "miracle of appropriateness" -- in various problems of
theoretical physics. The notable examples include the two-dimensional Ising
model, one-dimensional impenetrable Bose gas, corner and polynuclear growth
models, one dimensional directed polymers, string theory, two dimensional
quantum gravity, and spectral distributions of random matrices. In the present
contribution, ideas of integrability are utilized to advocate emergence of an
one-dimensional Toda Lattice and the fifth Painlev\'e transcendent in the
paradigmatic problem of conductance fluctuations in quantum chaotic cavities
coupled to the external world via ballistic point contacts. Specifically, the
cumulants of the Landauer conductance of a cavity with broken time-reversal
symmetry are proven to be furnished by the coefficients of a Taylor-expanded
Painlev\'e V function. Further, the relevance of the fifth Painlev\'e
transcendent for a closely related problem of sample-to-sample fluctuations of
the noise power is discussed. Finally, it is demonstrated that inclusion of
tunneling effects inherent in realistic point contacts does not destroy the
integrability: in this case, conductance fluctuations are shown to be governed
by a two-dimensional Toda Lattice.
