I liked the recent question from user goebeld and especially the answer from Rouben Rostamian.
I admit, I didn’t even realize that Maple had VariationalCalculus procedures.
But what if the red and green  points are on the surface x1^4 + x2^4 + x3^4 -1 = 0 ? 
Points coordinates (-0.759835685700000, -0.759835685700000, 0.759835685700000) and
 (0.759835685700000, 0.759835685700000, -0.759835685700000).

Where will the shortest distance between these points on a given surface be? Taking into account symmetry, of course.

It was found on the social networks of the WM group. Written in Python. Perhaps someone would like to adopt it.
 

The question is not at all from me, but, probably, one might say, from the authors of this publication.  interesting_system.pdf

Just for fun.
Find all real solutions to this 2x2 system of nonlinear equations in any given domain. 

f1:=x1-x1*sin(x1+5x2)-x2*cos(5x1-x2);
f2:=x2-x2*sin(5x1-3x2)+x1*cos(3x1+5x2);

Inscribed square problem

I decided to check on this curve 
 

 4*(x1-0.25)^4-x1^2*x2^2+(x2-0.25)^4-1.21=0;

I get a very rough solution, because the difference between the sides of the "square" begins already at 1-2 decimal places. More precisely, it doesn’t work, that is, we can say that I personally could not find confirmation of the hypothesis.
The coordinates of the vertices of the square:

                    -0.4823584672, -0.2770841741

                    0.9883885535, -0.3959790155

                    1.108267478, 1.086941264

                   -0.3459185869, 1.219514527

Side lengths:
                          1.475544911

                          1.487757882

                          1.460216690

                          1.502805215


 

Perhaps someone would like to try.

Another training example (number 2 and last) for finding all solutions to a system of equations:

f1 := x3^2-0.1*x1^4-0.05*x2^4+1;
f2 := x1^3+x2^3+0.05*x3^3-1; 
f3 := -2*cos(3*x1)+2*cos(3*x2)-2*cos(3*x3)+1;

In my version, there are 116 solutions.
Is it so?