Equi-Affine surfaces 

fr > coord_frame([x,y],[u],fr):
 

fr> LieAlg:=[[1,0,0],[0,1,0],[0,0,1],[y,0,0],[u[0,0],0,0],[0,x,0],[0,u[0,0],0],[0,0,x],[0,0,y],[x,0,-u[0,0]],[0,y,-u[0,0]]]:    
LieAlg:=map( V-> v_zip(V,frameJetVariables(fr,0)[1..3],plus, vect), LieAlg);
 

(Typesetting:-mprintslash)([LieAlg := [D_x[``], D_y[``], D_u[[0, 0]][``], y*D_x[``], u[0, 0]*D_x[``], x*D_y[``], u[0, 0]*D_y[``], x*D_u[[0, 0]][``], y*D_u[[0, 0]][``], x*D_x[``]-u[0, 0]*D_u[[0, 0]][``...
(Typesetting:-mprintslash)([LieAlg := [D_x[``], D_y[``], D_u[[0, 0]][``], y*D_x[``], u[0, 0]*D_x[``], x*D_y[``], u[0, 0]*D_y[``], x*D_u[[0, 0]][``], y*D_u[[0, 0]][``], x*D_x[``]-u[0, 0]*D_u[[0, 0]][``...
(Typesetting:-mprintslash)([LieAlg := [D_x[``], D_y[``], D_u[[0, 0]][``], y*D_x[``], u[0, 0]*D_x[``], x*D_y[``], u[0, 0]*D_y[``], x*D_u[[0, 0]][``], y*D_u[[0, 0]][``], x*D_x[``]-u[0, 0]*D_u[[0, 0]][``... 

Hyperbolic section 

Elliptic section