This is the easiest program text of the method, which I could do. I think someone wants to deal with the idea of the method, the text will help him and it will be easier to deal with other examples of the application and then create their own programs.

@Axel Vogt       
     Thank you for your attention to the subject and for your constructive, as I understood  the message. If Google translator has translated all right on both sides, then:
     Yes, 2d and 3d convenient to use implicitplot  to see, but what is the accuracy and computation time? And how to make the calculation of motion mechanisms when the dimension, for example, > 12d?  As for the abstract shapes in the form of a complex set of solutions and higher dimensions, we can move in very small increments (h) and to monitor the value of abs (F (X)) <eps, and create a set of points, constantly changing direction, or to pick up auxiliary equation. When we seek solutions by this method, it is sufficient to have a single point on the set of solutions that from it very quickly and accurately obtain an infinite number of other points. We do not iterate over all grid points in some region of space where the supposed existence of a solution but move on to the set of solutions.
     My example  "bublik" is just an illustration, but examples the mechanisms it is work that is very difficult to perform using implicitplot (or rather impossible), but using this method mechanisms are calculated quickly and easily.  
     In my posts given examples of geodesic computations. Geodesics calculated numerically in any direction and at any distance on any piecewise smooth surfaces. There, too, the proposed method is applied.

Work of this method with examples of mechanisms with two or more degrees of freedom.
Made in SMath by Friedel Selitsky.

http://en.smath.info/forum/yaf_postst2734findunread_Animation-of-mechanisms-with-two-or-more-degrees-of-freedom.aspx#post14844  

@Markiyan Hirnyk  Hirnyk, Spring Fever instead of absorbing knowledge?

@Markiyan Hirnyk  Hirnyk, read and learn the definition of "Connected space". At your age, it is high time to know. After that, Hirnyk, if you are a normal person, you apologize for your latest folly.

 fsolve  not a magic wand. As far as  I know,  there  is involved  the Newton method. By modern  standards   it is a very  poor method –  this method uses a  matrix of partial derivatives. Try,  for  example,  DirectSearch.

https://vk.com/doc242471809_363265138

https://vk.com/doc242471809_364348806
https://vk.com/doc242471809_364348545  

https://vk.com/doc242471809_360806878
https://vk.com/doc242471809_361267038
https://vk.com/doc242471809_361540922

@Markiyan Hirnyk In order to you were not bored, try something like this. As a hint showing the surface equation and remind you that the motion on the surface in any space can be not only a picture, but at the same time, the solution of the corresponding equation.
x3-0.1e-1*exp(x1)/(0.94e-2+x1^4+x2^4)=0;  
x3-.1*(sin(4*x1)+sin(3*x2+x3)+sin(2*x2))=0;


https://vk.com/doc242471809_361089459
https://vk.com/doc242471809_360840447

Additional examples  on other surfaces

     Your system is one equation for two variables x1, x2:

eq: = tanh (x1) + cosh (x1 + x2) - 2.

     Any solution to this system is a solution of the original system. Choose from all what you need.

o@Axel Vogt 

       Consider the simplest situation. Numerical method in real space. Points of the curve are substituted in the plane equation P (x) = 0. If abs (P (x)) <= delt, the point belongs to a plane. The equation of the plane is made by the first three points of curve that allow you to obtain the equation of the plane. Accordingly, if N (the dimension of the space R ^N) > 3, then for the equation of the plane should be taken  > 3 points of the curve.  The curve can be set or separate equations coordinates of the points of the curve, or by solving a system of nonlinear equations N-1 to the number of variables N, polynomial or transcendental equations.
       Sorry, my English through a Google translator.

@Markiyan Hirnyk 

      For all points of the curve is considered one single equation of the plane. "Any three points" - are any suitable first three points on the curve.

Can use any other suitable method.

@Markiyan Hirnyk 

      Hirnyk, my best friend. Let's proceed as follows. Since it is necessary to compare the comparable, make yourself at least something on the proposed directions. Show that you can to do it. And Use package Sergei Moiseyev I can too, but do it only if necessary. You were asked at forum Exponenta.ru to learn how to solve the ODE with boundary conditions by using package of Sergei Moiseyev . You pretended that not to notice the proposal. And that now rend the air?
      Yes, relax and learn to write real programs.