  The present paper studies a completely integrable conformally invariant model
in 1+1 dimensions that corresponds to string propagation on the two-dimensional
black hole background (semi-ininite cigar). Besides the two space-time string
fields there is a third (internal) field with a very specific Liouville-type
interaction leading to the complete integrability. This system is known as
non-abelian Toda theory. I give the general explicit classical solution. It
realizes a rather involved transformation expressing the interacting string
fields in terms of (three) functions $\varphi_j(u)$ and $\bar\varphi_j(v)$ of
one light-cone variable only. The latter are shown to lead to standard harmonic
oscillator (free field) Poisson brackets thus paving the way towards
quantization. There are three left-moving and three right-moving conserved
quantities. The right (left)-moving conserved quantities form a new closed
non-linear, non-local Poisson bracket algebra. This algebra is a Virasoro
algebra extended by two conformal dimension-two primaries.
