  We develop a constructive procedure for arriving at the Hamilton-Jacobi
framework for the so-called affine in acceleration theories by analysing the
canonical constraint structure. We find two scenarios in dependence of the
order of the emerging equations of motion. By properly defining generalized
brackets, the non-involutive constraints that originally arose, in both
scenarios, may be removed so that the resulting involutive Hamiltonian
constraints ensure integrability of the theories and, at the same time, lead to
the right dynamics in the reduced phase space. In particular, when we have
second-order in derivatives equations of motion we are able to detect the gauge
invariant sector of the theory by using a suitable approach based on the
projection of the Hamiltonians onto the tangential and normal directions of the
congruence of curves in the configuration space. Regarding this, we also
explore the generators of canonical and gauge transformations of these
theories. Further, we briefly outline how to determine the Hamilton principal
function $S$ for some particular setups. We apply our findings to some
representative theories: a Chern-Simons-like theory in $(2+1)$-dim, an harmonic
oscillator in $2D$ and, the geodetic brane cosmology emerging in the context of
extra dimensions.
