  The presence and character of local integrals of motion -- quasi-local
operators that commute with the Hamiltonian -- encode valuable information
about the dynamics of a quantum system. In particular, strongly disordered
many-body systems can generically avoid thermalisation when there are
extensively many such operators. In this work, we explicitly construct local
conserved operators in $1$D spin chains by directly minimising their commutator
with the Hamiltonian. We demonstrate the existence of an extensively large set
of local integrals of motion in the many-body localised phase of the disordered
XXZ spin chain. These operators are shown to have exponentially decaying tails,
in contrast to the ergodic phase where the decay is (at best) polynomial in the
size of the subsystem. We study the algebraic properties of localised
operators, and confirm that in the many-body localised phase they are
well-described by "dressed" spin operators.
