  We introduce the most general version of Dubrovin-type equations for divisors
on a hyperelliptic curve of arbitrary genus, and provide a new argument for
linearizing the corresponding completely integrable flows. Detailed
applications to completely integrable systems, including the KdV, AKNS, Toda,
and the combined sine-Gordon and mKdV hierarchies, are made. These
investigations uncover a new principle for 1+1-dimensional integrable soliton
equations in the sense that the Dubrovin equations, combined with appropriate
trace formulas, encode all hierarchies of soliton equations associated with
hyperelliptic curves. In other words, completely integable hierarchies of
soliton equations determine Dubrovin equations and associated trace formulas
and, vice versa, Dubrovin-type equations combined with trace formulas permit
the construction of hierarchies of soliton equations.
