The sym_implicit subroutine of the odeadvisor command tests if a given first order ODE in "implicit form" (that is, dy/dx cannot be isolated) has one or more of the following symmetries:

[xi=0, eta=y], [xi=0, eta=x], [xi=0, eta=1/x], [xi=0, eta=1/y],
[xi=x, eta=0], [xi=y, eta=0], [xi=1/x, eta=0], [xi=1/y, eta=0],
[xi=x, eta=y], [xi=y, eta=x]:

where the infinitesimal symmetry generator is given by the following:

G := f -> xi*diff(f,x) + eta*diff(f,y);

This routine is relevant when using symmetry methods for solving high-degree ODEs (non linear in dy/dx). The cases [xi=0, eta=y], [xi=1/x, eta=y] and [xi=x, eta=y] cover the families of homogeneous ODEs mentioned in Murphy's book, pages 63-64.