F := m*z^4-4*m*z^3+(3*m+3)*z^2-6*u;
 allvalues(solve(F, z));
...and further select intervals.

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.)