[Mu] = 0.01214398779977472`

[Mu]1 = 328823.8592475095`

 Subscript[e, 1] = 0.0167`

  A = 389.1779396462019`

  [Phi] = 0

  Subscript[e, 2] = 0.0549`
  Vx = x + [Mu]/(y^2 + (-1 + x + [Mu])^2)^(3/2) - (
   x [Mu])/(y^2 + (-1 + x + [Mu])^2)^(
   3/2) - [Mu]^2/(y^2 + (-1 + x + [Mu])^2)^(3/2) -
   x/(y^2 + (x + [Mu])^2)^(3/2) - [Mu]/(y^2 + (x + [Mu])^2)^(
   3/2) + (x [Mu])/(y^2 + (x + [Mu])^2)^(
   3/2) + [Mu]^2/(y^2 + (x + [Mu])^2)^(3/2) + (x [Mu]1)/(2 A^3) + (
   3 x [Mu]1 Cos[2 [Phi]])/A^3 + (3 y [Mu]1 Sin[2 [Phi]])/
   A^3 - ([Mu]1 Cos[[Phi]]
   !(*SubsuperscriptBox[(e), (1), (2)]))/(2 A^2);

  Vy = y - (y [Mu])/(y^2 + (-1 + x + [Mu])^2)^(3/2) -
  y/(y^2 + (x + [Mu])^2)^(3/2) + (y [Mu])/(y^2 + (x + [Mu])^2)^(
  3/2) + (y [Mu]1)/(2 A^3) - (3 y [Mu]1 Cos[2 [Phi]])/
  A^3 + ([Mu]1 Sin[[Phi]])/A^2 + (3 x [Mu]1 Sin[2 [Phi]])/
  A^3 - ([Mu]1 Sin[[Phi]]
  !(*SubsuperscriptBox[(e), (1), (2)]))/(2 A^2);

 Vxx = D[Vx, x];
 Vyy = D[Vy, y];
 Vxy = D[Vx, y];
 Vyx = D[Vy, x];

 newton[{x_, y_}] := {x, y} + {Simplify[-(Vx Vyy - Vy Vxy)/(Vyy Vxx -
   Vxy^2)],  Simplify[(Vx Vyx - Vy Vxx)/(Vyy Vxx - Vxy^2)]};

 sol = {{ 0.188277, -0.9824026}, { 0.188277, 0.9824026}, { 1.152706, 0},
 {-0.998550, 0}, { 0.987856, 0}, { 0.835518, 0}, { -0.012144, 0}};
 tab = ParallelTable[ N[FixedPoint[newton, {i, j}, 10]], {j, -2, 2, 0.003},
     {i, -2, 2, 0.003}]

 rules = Rule @@@ Transpose[{sol, Range[Length[sol]]}];

 newtab = Map[First@Nearest[rules, #] &, tab, {2}];

 P1 = ArrayPlot[newtab, ColorFunction -> "Rainbow",  DataReversed -> True]