  We construct two infinite families of algebraic minimal cones in $R^{n}$. The
first family consists of minimal cubics given explicitly in terms of the
Clifford systems. We show that the classes of congruent minimal cubics are in
one to one correspondence with those of geometrically equivalent Clifford
systems. As a byproduct, we prove that for any $n\ge4$, $n\ne 16k+1$, there is
at least one minimal cone in $R^{n}$ given by an irreducible homogeneous cubic
polynomial. The second family consists of minimal cones in $R^{m^2}$, $m\ge2$,
defined by an irreducible homogeneous polynomial of degree $m$. These examples
provide particular answers to the questions on algebraic minimal cones posed by
Wu-Yi Hsiang in the 1960's.
