  We examine the general element of the class of ordinary differential
equations, $yy^{(n+1)}+\alpha y'y^{(n)}=0$, for its Lie point symmetries. We
observe that the algebraic properties of this class of equations display an
attractive set of patterns, the general member of the class can have three type
of Algebra, $(n+1)A_1 \oplus_s\{A_1 \oplus sl(2,R)\}$, $A_1 \oplus sl(2,R)$ or
$A_2 \oplus A_1$, for different values of $\alpha$. We look at the singularity
properties of these equations for various values of $\alpha$.
