In this paper, we study an element which is both group invertible and Moore Penrose invertible to be EP, partial isometry and strongly EP by discussing the existence of solutions in a definite set of some given constructive equations. Mainly, let a ∈ R # ∩ R +. Then we firstly show that an element a ∈ R EP if and only if and Equation : axa + + a + ax = 2x has at least one solution in χ a = {a, a # , a + , a * , (a #) * , (a +) * }. Next, a ∈ R SEP if and only if Equation: axa * + a + ax = 2x has at least one solution in χ a. Finally, a ∈ R PI if and only if Equation: aya * x = xy has at least one solution in ρ 2 a , where ρ a = {a, a # , a + , (a #) * , (a +) * }