It is often convenient to represent probabilistic models in a first-order fashion, using logical atoms such as $partners(X,Y)$ as random variables parameterized by logical variables. de Salvo Braz, Amir and Roth, following(Poole, give a lifted variable elimination algorithm (FOVE) for computing marginal probabilities from first-order probabilistic models (belief assessment, or BA). FOVE is lifted because it works directly at the first-order level, eliminating all the instantiations of a set of atoms in a single step, in some cases independently of the number of these instantiations. Previous work could treat only restricted potential functions. There, atoms' instantiations cannot constrain each other: predicates can appear at most once, or logical variables must not interact across atoms. In this paper, we present two contributions. The first one is a significantly more general lifted variable elimination algorithm, FOVE-P, that covers many cases where atoms share logical variables. The second contribution is to use FOVE-P for solving the Most Probable Explanation (MPE) problem, which consists of calculating the most probable assignment of the random variables in a model. The transition from BA to MPE is straightforward in propositional models, but the lifted first-order case is harder. We introduce the notion of lifted assignments, a distribution of values to a set of random variables rather than to each individual one. Lifted assignments are cheaper to compute while being as useful as regular assignments over that group. Both contributions advance the theoretical understanding of lifted probabilistic inference.