  Quantum many-body scars are highly excited eigenstates of non-integrable
Hamiltonians which violate the eigenstate thermalization hypothesis and are
embedded in a sea of thermal eigenstates. We provide a general mechanism to
construct partially integrable models with arbitrarily large local Hilbert
space dimensions, which host exact many-body scars. We introduce designed
integrability-breaking terms to several exactly solvable spin chains, whose
integrable Hamiltonians are composed of the generators of the Temperley-Lieb
algebra. In the non-integrable subspace of these models, we identify a special
kind of product states -- the generalized spin helix states as exact quantum
many-body scars, which lie in the common null space of the non-Hermitian
generators of the Temperley-Lieb algebra and are annihilated by the
integrability-breaking terms. Our constructions establish an intriguing
connection between integrability and quantum many-body scars, meanwhile provide
a systematic understanding of scarred Hamiltonians from the perspective of
non-Hermitian projectors.
