The problem arose while playing with inscribed circles, as in this post. If anyone is interested, try to find a circle of maximum radius inscribed between the curves
x1^2 + 2*x2^2 - 1 = 0 and (x1 - sin(x1))^2 + (x2 - sin(x2))^2 - 1 = 0.

Curve graphs.

A little continuation of topics 1 and 2. This is a very similar cube from 2, 
 

but with a different equation:

f1 := (x1-sin(x1))^2+(x2-sin(x2))^2+(x3-sin(x3))^2-0.02513144866;
And other point coordinates (-.8283302152, -.8283302152, .8283302152) and (.8283302152, .8283302152, -.8283302152).

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);