  We present numerical evidence that solutions of the Bethe Ansatz equations
for a Bloch particle in an incommensurate magnetic field (Azbel-Hofstadter or
AH model), consist of complexes-"strings". String solutions are well-known from
integrable field theories. They become asymptotically exact in the
thermodynamic limit. The string solutions for the AH model are exact in the
incommensurate limit, where the flux through the unit cell is an irrational
number in units of the elementary flux quantum.
  We introduce the notion of the integral spectral flow and conjecture a
hierarchical tree for the problem. The hierarchical tree describes the topology
of the singular continuous spectrum of the problem. We show that the string
content of a state is determined uniquely by the rate of the spectral flow
(Hall conductance) along the tree. We identify the Hall conductances with the
set of Takahashi-Suzuki numbers (the set of dimensions of the irreducible
representations of $U_q(sl_2)$ with definite parity).
  In this paper we consider the approximation of noninteracting strings. It
provides the gap distribution function, the mean scaling dimension for the
bandwidths and gives a very good approximation for some wave functions which
even captures their multifractal properties. However, it misses the
multifractal character of the spectrum.
