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My potential is the form as:

$$ U = 0.515755 (x^2 + y^2) - 1.0156333 x(0.0200/((-0.1 + x)^2 + y^2)^{(3/2)} + 0.006750/((0.9 + x)^2 + y^2)^{(7/2)} - 0.0180000/((0.9 + x)^2 + y^2)^{(5/2)} - 3.6/((0.9 + x)^2 + y^2)^{(3/2)} - 0.3600000/(((0.9 + x)^2 + y^2) \sqrt{0.0001 + (x^2 + y^2)^2})) + 1.0156333 (x^2 + y^2) (0.2/((-0.1 + x)^2 + y^2)^{(3/2)} - 0.0075/((0.9 + x)^2 + y^2)^{(7/2)} + 0.02/((0.9 + x)^2 + y^2)^{(5/2)} + 4/((0.9 + x)^2 + y^2)^{(3/2)} + 0.4`/(((0.9 + x)^2 + y^2) \sqrt{0.0001 + (x^2 + y^2)^2})) $$

In Mathematica code:

U[x_, y_]:= 
  0.515755 (x^2 + y^2) - 1.0156333 x (0.0200/((-0.1 + x)^2 + y^2)^( 3/2) + 
   0.006750/((0.9 + x)^2 + y^2)^(7/2) - 0.0180000/((0.9 + x)^2 + y^2)^(5/2) - 
   3.6/((0.9 + x)^2 + y^2)^(3/2) - 0.3600000/(((0.9 + x)^2 + y^2)*Sqrt[0.0001 + (x^2 + y^2)^2])) +
   1.0156333*(x^2 + y^2)*(0.2/((-0.1 + x)^2 + y^2)^(3/2) - 0.0075/((0.9 + x)^2 + y^2)^(7/2) +
   0.02/((0.9 + x)^2 + y^2)^( 5/2) + 4/((0.9 + x)^2 + y^2)^(3/2) +
   0.4/(((0.9 + x)^2 + y^2)*Sqrt[0.0001` + (x^2 + y^2)^2])
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    $\begingroup$ What have you tried? $\endgroup$ – bbgodfrey Mar 15 '20 at 14:20
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    $\begingroup$ Welcome to Mathematica.SE! I hope you will become a regular contributor. To get started, 1) take the introductory tour now, 2) when you see good questions and answers, vote them up by clicking the gray triangles, because the credibility of the system is based on the reputation gained by users sharing their knowledge, 3) remember to accept the answer, if any, that solves your problem, by clicking the checkmark sign, and 4) give help too, by answering questions in your areas of expertise. $\endgroup$ – bbgodfrey Mar 15 '20 at 14:22
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your Mathematica code has some syntax error I corrected it. This is what I have done:

1) eqn

 U[x_, y_] :=  0.515755 (x^2 + y^2) - 1.0156333 x (0.0200/((-0.1 + x)^2 + y^2)^(3/2) + 
 0.006750/((0.9 + x)^2 + y^2)^(7/2) - 
 0.0180000/((0.9 + x)^2 + y^2)^(5/2) - 
 3.6/((0.9 + x)^2 + y^2)^(3/2) - 
 0.3600000/(((0.9 + x)^2 + y^2)*Sqrt[0.0001 + (x^2 + y^2)^2])) + +1.0156333*(x^2 + y^2) (0.2/((-0.1 + x)^2 + y^2)^(3/2) - 
 0.0075/((0.9 + x)^2 + y^2)^(7/2) + 
 0.02/((0.9 + x)^2 + y^2)^(5/2) + 4/((0.9 + x)^2 + y^2)^(3/2) + 
 0.4/(((0.9 + x)^2 + y^2)*Sqrt[0.0001` + (x^2 + y^2)^2]));

2) Take partial Derivatives to find functions to minimize

   funs = D[U[x, y], #] & /@ {x, y};

3) Generally, you equate these to zero and solve it for variables but for a complex equation often we resort to numerical methods. You can use FindAllCrossings2D for this purpose but you need to know ranges of x and y's:

   pts = FindAllCrossings2D[funs, {x, -1, 1}, {y, -1, 1}, Method -> 
        {"Newton", "StepControl" -> "LineSearch"},  PlotPoints -> 85]//Chop

Just do a ListPlot of these to find the plot of x and y values.

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