Algebro-geometric integration of the Boussinesq hierarchy

We construct an integrable hierarchy of the Boussinesq equation using the Lie-algebraic approach of Holod-Flashka-Newell-Ratiu. We show that finite-gap hamiltonian systems of the hierarchy arise on coadjoint orbits in the loop algebra of $\mathfrak{sl}(3)$, and possess spectral curves from the family of $(3,3N\,{+}\,1)$-curves, $N\,{\in}\, \Natural$. Separation of variables leads to the Jacobi inversion problem on the mentioned curves, which is solved in terms of the corresponding multiply periodic functions. An exact finite-gap solution of the Boussinesq equation is obtained explicitly, and a conjecture on the reality conditions is made. The obtained solutions are computed for several spectral curves, and illustrated graphically.