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I have to plot, in the plane, the following equation

$$ (x_1+x_2-2)^6=64(2x_2-3x_1+1),\quad x_1\leq 0 $$ $$ (x_1+x_2+2)^6=64(-2x_2+3x_1+1),\quad x_1\geq 0.$$

Maybe there is a more appropriate function to plot this but I managed something with ContourPlot:

Show[
 {
  ContourPlot[{(x1 + x2 - 2)^6 == 64 (2 x2 - 3 x1 + 1)}, {x1, -8, 
    8}, {x2, -8, 8}, GridLines -> Automatic, 
   ContourStyle -> {Black, Dashed}],
  ContourPlot[{(x1 + x2 + 2)^6 == 64 (-2 x2 + 3 x1 + 1)}, {x1, -8, 
    8}, {x2, -8, 8}, GridLines -> Automatic, 
   ContourStyle -> {Red, Dashed}]
  }
 ]

Which gives enter image description here

However, for the black-dashed curve, I only need the lower portion (for which $x_1\leq 0$) and for the red-dashed curve, I only need the upper portion (for which $x_1\geq 0$).

I tried using the RegionFunction option but I could only eliminate a portion of what needed to be taken off:

enter image description here

How can I manage this?

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Another way, solving for an explicit formula for the desired function:

Plot[
 x2 /. {First@
     Solve[(x1 + x2 - 2)^6 == 64 (2 x2 - 3 x1 + 1) && x1 <= 0, x2, 
      Reals],
    Last@Solve[(x1 + x2 + 2)^6 == 64 (-2 x2 + 3 x1 + 1) && x1 >= 0, 
      x2, Reals]} // Evaluate,
 {x1, -8, 8},
 PlotStyle -> {Directive[Black, Dashed], Directive[Red, Dashed]},
 PlotRange -> 8, Frame -> True, GridLines -> Automatic, 
 AspectRatio -> 1
 ]
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I think your issue might be that $x_1 \leq 0$ and $x_1 \geq 0$ do not properly characterize the parts of the regions you want. It seems that for the bottom one, you want $x_1 \geq -x_2$, and for the top you want $x_1 \leq -x_2$. Then RegionFunction works:

Show[{ContourPlot[{(x1 + x2 - 2)^6 == 64 (2 x2 - 3 x1 + 1)}, {x1, -8, 
    8}, {x2, -8, 8}, GridLines -> Automatic, 
   ContourStyle -> {Black, Dashed}, RegionFunction -> (#1 <= -#2 &)], 
  ContourPlot[{(x1 + x2 + 2)^6 == 64 (-2 x2 + 3 x1 + 1)}, {x1, -8, 
    8}, {x2, -8, 8}, GridLines -> Automatic, 
   ContourStyle -> {Red, Dashed}, RegionFunction -> (#1 >= -#2 &)]}]

enter image description here

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I don't know how to translate the contour to a parametric curve directly. So here we try to differential the equation then we get a differential equation and solve it by NDSolve.

$$\frac{dx2}{dx1}=-\frac{\partial{f(x1,x2)}}{\partial x1}/\frac{\partial{f(x1,x2)}}{\partial x2}$$

f[x1_, x2_] := (x1 + x2 - 2)^6 - 64 (2 x2 - 3 x1 + 1);
solf = NDSolve[{D[yf[t], t] D[f[x1, x2], x2] == -D[f[x1, x2], 
        x1] /. {x2 -> yf[t], x1 -> t}, yf[0] == 0}, yf, {t, -10, 0}];
g[x1_, x2_] := (x1 + x2 + 2)^6 - 64 (-2 x2 + 3 x1 + 1);
solg = NDSolve[{D[yg[t], t] D[g[x1, x2], x2] == -D[g[x1, x2], 
        x1] /. {x2 -> yg[t], x1 -> t}, yg[0] == 0}, yg, {t, 0, 10}];
Show[Plot[yf[t] /. solf, {t, -6, 0}, GridLines -> Automatic, 
  PlotStyle -> {Black, Dashed}], 
 Plot[yg[t] /. solg, {t, 0, 6}, PlotStyle -> {Red, Dashed}], 
 PlotRange -> All]

enter image description here

Edit

We can get the parametric curve by differential the equation and set the curve with unit speed velocity.

$$df=\frac{\partial{f}}{\partial x_1}dx_1+\frac{\partial{f}}{\partial x_2}dx_2$$ $$ x_{1}'(t)^2+x_{2}'(t)^2=1 $$

f[x1_, x2_] := (x1 + x2 - 2)^6 - 64 (2 x2 - 3 x1 + 1);
df = Dt[f[x1, x2]] /. {x1 -> x1[t], x2 -> x2[t]} /. Dt[t] -> 1;
solf = NDSolve[{df == 0, x1'[t] + x2'[t] == 1, x1[0] == 0, 
    x2[0] == 0}, {x1, x2}, {t, -6, 6}];
g[x1_, x2_] := (x1 + x2 + 2)^6 - 64 (-2 x2 + 3 x1 + 1);
dg = Dt[g[x1, x2]] /. {x1 -> x1[t], x2 -> x2[t]} /. Dt[t] -> 1;
solg = NDSolve[{dg == 0, x1'[t] + x2'[t] == 1, x1[0] == 0, 
    x2[0] == 0}, {x1, x2}, {t, -6, 6}];
Show[ParametricPlot[{x1[t], x2[t]} /. solf, {t, -1, 2}, 
  PlotStyle -> {Dashed}, Mesh -> {{0}}, 
  MeshShading -> {Black, Green}], 
 ParametricPlot[{x1[t], x2[t]} /. solg, {t, -2, 1}, 
  PlotStyle -> {Dashed}, Mesh -> {{0}}, MeshShading -> {Cyan, Red}], 
 PlotRange -> All, GridLines -> Automatic]

enter image description here

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You can use a single ContourPlot with region function Sign[#2] (# + #2) <= 0 & to restrict the plot to the region below the main diagonal for positive y and above the main diagonal for negative y values:

ContourPlot[{(x1 + x2 - 2)^6 == 64 (2 x2 - 3 x1 + 1), 
   (x1 + x2 + 2)^6 ==  64 (-2 x2 + 3 x1 + 1)}, 
 {x1, -8, 8}, {x2, -8, 8}, 
 GridLines -> Automatic, 
 ContourStyle -> Thread[{Dashed, {Black, Red}}], 
 RegionFunction -> (Sign[#2] (# + #2) <= 0 &)]

enter image description here

You can also use

ContourPlot[{ConditionalExpression[(x1 + x2 - 2)^6 - 64 (2 x2 - 3 x1 + 1), 
     Sign[x2] (x1 + x2) <= 0], 
  ConditionalExpression[(x1 + x2 + 2)^6 - 64 (-2 x2 + 3 x1 + 1), 
     Sign[x2] (x1 + x2) <= 0]},
 {x1, -8, 8}, {x2, -8, 8},
 GridLines -> Automatic, 
 ContourStyle -> Thread[{Dashed, {Black, Red}}]]
same picture
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