  This paper presents a method of computing approximate conservation laws and
eigenstates of integrability-broken models using the concept of adiabatic
continuation. Given some Hamiltonian, eigenstates and conserved operators may
be computed by using those of a simple Hamiltonian close by in parameter space,
dressed by some unitary rotation. However, most adiabatic continuation analyses
only use this unitary implicitly. In this work, approximate adiabatic gauge
potentials are used to construct a state dressing using variational methods, to
compute eigenstates via a rotated truncated spectrum approximation. These
methods allow construction of both low and high-energy approximate nonthermal
eigenstates, as well as quasi-local almost-conserved operators, in models where
integrability may be non-perturbatively broken. These concepts will be
demonstrated in the mixed-field Ising model.
