  We classify the soliton data in the totally non--negative part of Gr(k,n)
which may be associated to algebraic-geometric data on certain rational
degenerations of regular hyperelliptic M-curves. Such degenerate rational curve
G is a desingularization of that constructed in Abenda-Grinevich
(arXiv:1506.00563) for soliton data in the totally positive part of Gr(n-1,n)
and the KP wavefunctions are the same in such case. G is also the curve
constructed in the paper for soliton data in the totally positive part of
Gr(1,n). For any such G and for any fixed k between 2 and (n-2), we show that
k-compatible soliton data correspond to a family of KP multi-line solitons
(T-hyperelliptic) which parametrize soliton data in an (n-1)--dimensional
variety of the totally positive part of the Grassmannian Gr(k,n). We explicitly
characterize T-hyperelliptic solitons from the algebraic-point of view.
T-hyperelliptic solitons are also connected to the solutions of the finite
non-periodic Toda hierarchy because the tau function is the same for both
systems. We investigate such relation from the algebraic point of view and
compare the two spectral problems. In particular, the vacuum KP divisor and the
Toda divisor coincide, while k--compatible divisors may be recursively
constructed using known recursions for the Toda system. Finally, we also
explain how KP divisors change under the space--time transformation which
induces a duality tranformation from soliton data in Gr(k,n) to soliton data in
Gr(n-k,n) and compare the action of such transformation both for the KP and the
Toda systems.
