The Draghilev method to find all the points of zero and Pi/2 slope of an implicit function f(x1,x2)=0 while driving along the section of the line connected.

Example solution of equation and searching of all the points of zero and Pi/2 slope (with animation):

x1^3+x2^3-0.1e-1*sin(1.00001*x1+x2)=0;

D__LIST0.mw

The Draghilev method for F(x,y,z)=0. Many points with good precision (variants with h and h1).

diffanimationAA.mw 

restart; with(RootFinding):

 f := Z^6-3*Z^4+3*Z^2+Z-1;

 rhs(op(1, Isolate(f, Z)));

 rhs(op(2, Isolate(f, Z)));

restart; with(RootFinding):

 Z := x+I*y;

 f := Z^6-3*Z^4+3*Z^2+Z-1;

 Isolate([evalc(Re(f)), evalc(Im(f))], [x, y]);

It seems that the system is not consistent in any combination of 12 equations…

 restart:

 g := (3*x^2-3*y^2)/(x-y);

 x := `5`; y := `2`;

 g;

MECAN1.mw 

       At the urging of Markiyan Alexeyevich I show the text. Because I can't find out the proper code at once, here is a very close substitute. The Draghilev method is used in my animation.

Only for real roots...

 restart;

 with(RootFinding):

eq1 := x^4+(2*(3*y+1))*x^2+(5*y^2+4*y+11)*x-y^2+10*y+2;

eq2 := y^3+(x-2)*y+x^2+x+2;

 Isolate([eq1, eq2], [x, y]);