  A formalism of systematic construction of integrable non-autonomous
deformations of soliton hierarchies is presented. The theory is formulated as
an initial value problem (IVP) for an associated Frobenius integrability
condition on a Lie algebra. It is showed that this IVP has a formal solution
and within the framewrok of two particular subalgebras of the hereditary Lie
algebra the explicit forms of this formal solution are presented. Finally, this
formalism is applied to Korteveg-de Vries, dispersive water waves and
Ablowitz-Kaup-Newell-Segur soliton hierarchies.
