@C_R 
Yes, the second question touches upon the strongest side of the Method. For example, in that very Schatz mechanism, one equation was discarded from the system of equations describing geometric connections, and we obtained a motion with small deformations. To generalize, we can find the infinite subset of solutions. After all, the set of solutions can break down into disconnected subsets (branches), and we need to find each branch. But it also happens that the entire set of solutions is connected, and in this case we solve the system at once, that is, in this case there is only one "pink" line. But we always need one starting point on the pink line, because this pink line is the solution to the Cauchy problem.
As for the first question (finding point solutions of the NxN system), the Method cannot guarantee a complete solution head-on, there are just successful examples. It is necessary to try to work with the system of equations in parts, as if working with a part of its equations. But even applying the Method head-on to NxN systems has undeniable advantages, the only thing, as it seems to me, is that it may be inferior to modern optimization methods for individual solutions. Therefore, optimization methods or Gröbner bases for polynomial systems of small dimension are very convenient for finding the starting point for pink lines.

@dharr  Yes, I also think that the successful solution of the original transcendental system  using polynomial theory is a fortunate coincidence of many circumstances.
Thanks again.

@dharr With your permission, I moved your comment to the answers, and +.
Thank you, there is something to learn about working with Maple.

@vv Yes, I used the Groebner package and figured out plex, but these were very simple and small examples. My PC got to your nops(G) in a reasonable time, but I'm a bit scared to try to go further.
Agree that it is possible to bring the work to the solution of a polynomial system only if there is a very serious interest. You have satisfied my curiosity completely.
If you are interested, at one time (back in the era of large computers) a group of several people was created for this task. True, I don’t know anything about the results.

@vv  Very well, +. Now we know that the polynomial interpretation also has solutions. (It is clear that a finite set of solutions of a polynomial system corresponds to a countable set of solutions of the original system.) Wonder how long it will take to find these solutions...

@C_R 
In my opinion, waiting so long is already a lot of work. Yes, I have long had suspicions about the real possibilities of solving polynomial systems on a theoretical basis. 
By the way, Draghilev's method allows us to draw conclusions about the solution of the original system itself, we can even say, to obtain its complete solution: for example, you can solve 4 of 5 equations in turn, tracking the change in the sign of the remaining equation.
In any case, I'm sure this is a very good example for Maple.

@janhardo 
maybe this is a call for everyone to use the search function on the forum more actively.

There was one activist here (Markiyan Hirnyk), so in order to protect myself from his nastiness, I turned to the community for help. But now he visits other forums.
(For some time now, I have also formally been a moderator, and, just in case, I will say right away that I have nothing to do with deleting non-advertising posts, that is, I have never deleted your posts.)

@Carl Love   I read both your messages in the thread with great pleasure. It seems to me that the online translator conveys the level of your mastery of the word.

@vv  No, it's not compact, it's professional☺.

@vv  

Thank you for your participation. Yes, it can be proved both geometrically and analytically.
This example was made specifically because of the animation. (I really like moving pictures.)

@C_R  
Well, you can try to evaluate the quality of the translation on one text from the MapleSoft application center. By the way, there is also a description of the Draghilev method. Translation from Russian to English online Google translator.

@C_R 
In other words, you don't know about the existence of online translators?

@sija 
 

I will show you my example with the rolling of the original ellipsoid
x1^2 + 5*x2^2 + 4*x3^2 - 0.25 = 0
on the transcendental surface
x3 = - 0.25*x1^2*sin(x1) - 0.25*sin(1.5*x2).

Of course, the greater the accuracy, the more time is required. To get animation on an old computer, we choose the optimal option. The equation of the ellipsoid at each point is printed on the graph.
for_an_ellipsoid.mw

(I have many different examples on this topic on the forum. I don't provide text for every example, because it's too cumbersome, but I try to show the basic program.)

@C_R  
In this case sqrt(b[5] ^2+ b[6]^2) is for the working point to move uniformly along the circle and also to make it easier to get the integration interval. But this has nothing to do with Draghilev's method. If you want, I can give you a list of literature in Russian that preceded the method, well, the method itself in Russian.
The author died long ago, and I do this as a hobby. I am not a professional. But I repeat, I will answer any questions about the Method.