  The main result of this paper is a construction of solutions to the reverse
Yang-Mills-Higgs flow converging in the $C^\infty$ topology to a critical
point. The construction uses only the complex gauge group action, which leads
to an algebraic classification of the isomorphism classes of points in the
unstable set of a critical point in terms of a filtration of the underlying
Higgs bundle.
  Analysing the compatibility of this filtration with the
Harder-Narasimhan-Seshadri double filtration gives an algebraic criterion for
two critical points to be connected by a flow line. As an application, we can
use this to construct Hecke modifications of Higgs bundles via the
Yang-Mills-Higgs flow. When the Higgs field is zero (corresponding to the
Yang-Mills flow), this criterion has a geometric interpretation in terms of
secant varieties of the projectivisation of the underlying bundle inside the
unstable manifold of a critical point, which gives a precise description of
broken and unbroken flow lines connecting two critical points. For non-zero
Higgs field, at generic critical points the analogous interpretation involves
the secant varieties of the spectral curve of the Higgs bundle.
