Non-log liftable log del Pezzo surfaces of rank one in characteristic five

Building upon the classification by Lacini [arXiv:2005.14544], we determine the isomorphism classes of log del Pezzo surfaces of rank one over an algebraically closed field of characteristic five either which are not log liftable over the ring of Witt vectors or whose singularities are not feasible in characteristic zero. We also show that the Kawamata-Viehweg vanishing theorem for ample $\mathbb{Z}$-Weil divisors holds for log del Pezzo surfaces of rank one in characteristic five if those singularities are feasible in characteristic zero.