restart; 
S := plottools[sphere]([0, 0, 0], 2, color = RGB(-.5, .1, .1), transparency = .5, style = surface):
plots[display](S);

ROOTOF.mw It remains to exclude unwanted values sigma . ( And allvalues to RootOf)

restart;
with(plots):
f1 := exp(x*y):
plot3d(f1, x = 0 .. 1, y =0 .. 1):
p1 := %:
p3 := intersectplot(z-f1, y-1, x = 0 .. 1, y = 0 .. 1, z = .3 .. 2.5, thickness = 6, axes = normal, numpoints = 6000):
display(p1, p3, axes = boxed);

Share your experience. In the “settings” exactly as instructed, I replaced my picture on the move. I only, remember that I spent a little time on the choice of its size. And so it was very easy.
Perhaps you have too many colors and fine details?

restart:
a := 4;
b := 5;
convert(`a+b = 4+5 `, symbol) = a+b;

I do not know… in my opinion, the easiest and most reliable way
   restart;
   z := x+I*y;
  #exp(z) +1;
  f1 := evalc(Re(exp(z)+1));
  f2 := evalc(Im(exp(z)+1));
  fsolve({f1, f2});
  solve({f1, f2});

Check whether the f function  of p and  function of r.
For example,
> depends (f, r)

I think it was done with the help of Drawing Tools...

 And superpositions of the functions do not need?

FUNCTION.mw 

for_you.mw 

Without fsolve, it is a polynomial system of equations.

I use this record to my calculations:
> restart;
   f := (x-a)*(x-b);
   f1 := unapply(diff(f, x), x);
   f1(3);
   f := unapply(f, x);
   f(3)

2DOF_1.mw

 Trying to solve the problem on the basis of Draghilev's method

 

     There is a chance to get all the solutions on connected subsets of solutions. Perhaps this is one connected subset. Here's an example where three variables and one equation.
http://www.mapleprimes.com/posts/200684-------Method-For-Solving-Underdetermined
     Can do similarly.  After receiving a set of solutions, we choose the ones that satisfy constrains.

       We are talking about a transformation of coordinates? To do this, we can transform corresponding equations. An example of the program (it's small and simple) transformation equations for surface motion in space.
http://www.mapleprimes.com/posts/200578-3d-Model-Of-A-Cam-Mechanism-Animation

And the example of the rigid body motion with six degrees of freedom.

https://vk.com/doc242471809_383336706

(But the calculation of geometry, kinematics and mechanisms animations I produce universal method based on solving systems of nonlinear equations underdetermined.)

Parameterization.mw
Maple15.
Numerical parameterization with an auxiliary surface. Always works in the local case and sometimes “global”.
Obtain the curve of intersection of the two surfaces, and then move it in any direction along the surface. Numerate and remember the point on the curve in each of its position. You can get a MxN matrix and use it as the index parameters of the surface.
Examples in the form of animation.
https://vk.com/doc242471809_375450085
https://vk.com/doc242471809_375799361
https://vk.com/doc242471809_375837573
https://vk.com/doc242471809_375682684