  We derive exact analytical solutions of the GOY shell model of turbulence. In
the absence of forcing and viscosity we obtain closed form solutions in terms
of Jacobi elliptic functions. With three shells the model is integrable. In the
case of many shells, we derive exact recursion relations for the amplitudes of
the Jacobi functions relating the different shells and we obtain a Kolmogorov
solution in the limit of infinitely many shells. For the special case of six
and nine shells, these recursions relations are solved giving specific analytic
solutions. Some of these solutions are stable whereas others are unstable. All
our predictions are substantiated by numerical simulations of the GOY shell
model. From these simulations we also identify cases where the models exhibits
transitions to chaotic states lying on strange attractors or ergodic energy
surfaces.
