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I rearranged the structure of the question for better understanding, without changing the core information.

How to plot a surface given implicitly by 3 equations in 5 coordinate variables?

I have 3 implicit equations in 5 variables: f1(S, yh, yd, xh, J)=0, f2(S, yh, yd, xh, J)=0, and f3(S, yh, yd, xh, J)=0. These equations determine a 3D surface in S-yh-yd coordinate system. The equations are given below.

M = 1;
roo = 0.01;
xd = 3;

f1[S_, yh_, yd_, xh_, J_]:= 2 M (xh^2 - (yh + 1)^2)/(xh^2 + (yh + 1)^2)^2 + 
  2 S ((xh - xd)^2 - (yh + yd)^2)/((xh - xd)^2 + (yh + yd)^2)^2 + 
  J/(2 yh) == 0,

f2[S_, yh_, yd_, xh_, J_]:= M*xh (yh + 1)/(xh^2 + (yh + 1)^2)^2 + 
  S (xh - xd) (yh + yd)/((xh - xd)^2 + (yh + yd)^2)^2 == 0,

f3[S_, yh_, yd_, xh_, J_]:= J*Log[2 yh*J/roo] + 2 M (yh + 1)/(xh^2 + (yh + 1)^2) + 
  2 S (yh + yd)/((xh - xd)^2 + (yh + yd)^2) - (Log[2/roo] + 1) == 0,

where S, yh, yd, xh, J are Reals, yh>0, and yd>0.

I am trying to obtain the surface in the S-yh-yd coordinate system like the following image. How to plot it? enter image description here

What I have tried: For 1 equation in 3 variables: f(S, yh, yd) = 0, I can use ContourPlot3D to get the surface; For equations expressed in an explicit form: S = S(xh, J), yh = yh(xh, J), yd = yd(xh, J), I can use ParametricPlot3D. But I failed when I faced this issue introduced above. Please help me. Thanks a lot for your support!