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We study a model of Rydberg atoms in a nearest-neighbor Rydberg blockadedregime, introduced by Lesanovsky in Phys. Rev. Lett. 108, 105301 (2012). Thismany-body model (which has one parameter $z$) has an exactly known gappedliquid ground state, and two exactly known low-lying excitations. We discovertwo new exact low-lying eigenstates. We also discuss behavior of the model atsmall parameter $z$ and its proximity to an integrable model. Lastly, wediscuss connections between the Lesanovsky model at intermediate $z$ andso-called PXP model. The PXP model describes a recent experiment that observedunusual revivals from a charge density wave initial state, which are attributedto a set of many-body "scar states" which do not obey the eigenstatethermalization hypothesis. We discuss the possibility of scar states in theLesanovsky model and present two approximations for them.

New exact eigenstates in the Lesanovsky model, proximity to integrability and the PXP model, and approximate scar states