  We establish the algebraic origin of the following observations made
previously by the authors and coworkers: (i) A given integrable PDE in $1+1$
dimensions within the Zakharov-Shabat scheme related to a Lax pair can be cast
in two distinct, dual Hamiltonian formulations; (ii) Associated to each
formulation is a Poisson bracket and a phase space (which are not compatible in
the sense of Magri); (iii) Each matrix in the Lax pair satisfies a linear
Poisson algebra a la Sklyanin characterized by the {\it same} classical $r$
matrix. We develop the general concept of dual Lax pairs and dual Hamiltonian
formulation of an integrable field theory. We elucidate the origin of the
common $r$-matrix structure by tracing it back to a single Lie-Poisson bracket
on a suitable coadjoint orbit of the loop algebra ${\rm sl}(2,\CC) \otimes \CC
(\lambda, \lambda^{-1})$. The results are illustrated with the examples of the
nonlinear Schr\"odinger and Gerdjikov-Ivanov hierarchies.
