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In this paper, we focus on partial isometry elements and strongly EP elements on a ring. We construct characterizing equations such that an element which is both group invertible and MP-invertible, is a partial isometry element, or is strongly EP, exactly when these equations have a solution in a given set. In particular, an element a ∈ R# ∩ R† is a partial isometry element if and only if the equation x = x(a†)*a† has at least one solution in {a, a#, a†, a*, (a#)*, (a†)*}. An element a ∈ R#∩R† is a strongly EP element if and only if the equation (a†)*xa† = xa†a has at least one solution in {a, a#, a†, a*, (a#)*, (a†)*}. These characterizations extend many well-known results. [ABSTRACT FROM AUTHOR]