[Rho] = 1; [CurlyEpsilon] := 0.8 ; T := 10 [Pi]

curvE = ParametricPlot[{[Rho]/(1 - [CurlyEpsilon]^2) Cos[[Psi]],
[Rho]/Sqrt[1 - [CurlyEpsilon]^2] Sin[[Psi]]}, {[Psi], 0, 2 [Pi]}]

Animate[FindRoot[{[Psi] - [CurlyEpsilon] Sin[[Psi]] ==
      2 [Pi] t/T} /. [CurlyEpsilon] -> 0.8 /.
   T -> 10 [Pi], {[Psi], 0}], {t, 0, 10 [Pi]}]

Animate[Show[curvE,
  Graphics[{PointSize[Large], Red,
    Point[Dynamic[{[Rho]/(1 - [CurlyEpsilon]^2) Cos[
         FindRoot[{[Psi] - [CurlyEpsilon] Sin[[Psi]] ==
            2 [Pi] t/T}, {[Psi], 0}]], [Rho]/
         Sqrt[1 - [CurlyEpsilon]^2] Sin[
         FindRoot[{[Psi] - [CurlyEpsilon] Sin[[Psi]] ==
            2 [Pi] t/T}, {[Psi], 0}]]}]]}]], {t, 0, 10 [Pi],
  AppearanceElements -> None}]