  The problem of finding all minimal surfaces presented in parametric form as
polynomials of certain degree is discussed by many authors. It is known that
the classical Enneper surface is (up to position in space and homothety) the
only polynomial minimal surface of degree 3 in isothermal parameters. In higher
degrees the problem is quite more complicated. Here we find a general form for
the functions that generate a polynomial minimal surface of arbitrary degree
via the Weierstrass formula and prove that any polynomial minimal surface of
degree 5 in isothermal parameters may be considered as belonging to one of
three special families.
