Why is my code finding false equilibrium points?

Let's define a simple system of non-linear equations regarding magnetic dipoles:

Clear["Global`*"]; r1 = Sqrt[(x - 0.5)^2 + y^2]; r2 = Sqrt[(x + 0.5)^2 + y^2]; p1 = 1/r1^3 + λ/r2^3; q1 = 1/2*(1/r1^3 - λ/r2^3); Ax = -y*p1; Ay = x*p1 - q1; Ω = ω*(x*Ay - y*Ax) + ω^2/2*(x^2 + y^2); Ωx = D[Ω, x]; Ωy = D[Ω, y]; ω = 1; λ = 1; 

My goal is to solve the system of nonlinear equations $\Omega_x(x,y) = 0, \Omega_y(x,y) = 0$ and determine the total number of equilibrium points.The total number of the equilibrium points strongly depends on the value of $\lambda$ and it is equal to 3, 5 or 7.

For this purpose, I used the accepted (first) answer of this question. Note that all the other suggested answers also fail to properly work for this particular system of nonlinear equations.

Options[FindRoots2D] = {PlotPoints -> Automatic, MaxRecursion -> Automatic}; FindRoots2D[funcs_, {x_, a_, b_}, {y_, c_, d_}, opts___] := Module[{fZero, seeds, signs, fy}, fy = Compile[{x, y}, Evaluate[funcs[[2]]]]; fZero = Cases[Normal[ ContourPlot[ funcs[[1]] == 0, {x, a - (b - a)/97, b + (b - a)/103}, {y, c - (d - c)/98, d + (d - c)/102}, Evaluate[FilterRules[{opts}, Options[ContourPlot]]]]], Line[z_] :> z, Infinity]; seeds = Flatten[((signs = Sign[Apply[fy, #1, {1}]]; #1[[1 + Flatten[ Position[Rest[signs*RotateRight[signs]], -1]]]]) &) /@ fZero, 1]; If[seeds == {}, {}, Select[Union[({x, y} /. FindRoot[{funcs[[1]], funcs[[2]]}, {x, #1[[1]]}, {y, #1[[2]]}, Evaluate[FilterRules[{opts}, Options[FindRoot]]]] &) /@ seeds, SameTest -> (Norm[#1 - #2] < 10^(-6) &)], a <= #1[[1]] <= b && c <= #1[[2]] <= d &]]] 

So

pts = FindRoots2D[{Ωx, Ωy}, {x, -5, 5}, {y, -5, 5}]; ContourPlot[{Ωx == 0, Ωy == 0}, {x, -2, 2}, {y, -2, 2}, ContourShading -> False, PlotPoints -> 100, PerformanceGoal :> "Quality", Epilog -> {AbsolutePointSize[6], Red, Point@pts}] 

And the output is the following

As you can see, the code finds five equilibrium points, which is of course wrong! There should be only three equilibrium points located on the $x$ axis, all with $y = 0$.

My question: Is there an efficient way to improve the suggested piece of code so to obtain the correct number of equilibrium points? Of course, any other alternative method for obtaining the equilibrium points will be highly appreciated.

I use version 9.0 of MMA.