  The integrable structure, recently revealed in some classical problems of the
theory of functions in one complex variable, is discussed. Given a simply
connected domain in the complex plane, bounded by a simple analytic curve, we
consider the conformal mapping problem, the Dirichlet boundary problem, and to
the 2D inverse potential problem associated with the domain. A remarkable
family of real-valued functionals on the space of such domains is constructed.
Regarded as a function of infinitely many variables, which are properly defined
moments of the domain, any functional from the family gives a formal solution
to the problems listed above. These functions are shown to obey an infinite set
of dispersionless Hirota equations. This means that they are $\tau$-functions
of an integrable hierarchy. The hierarchy is identified with the dispersionless
limit of the 2D Toda lattice. In addition to our previous studies, we show
that, with a more general definition of the moments, this connection is not
specific for any particular solution to the Hirota equations but reflects the
structure of the hierarchy itself.
