The problem of finding all minimal surfaces presented in parametric form
as polynomials 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.

Key Words: Minimal surface, isothermal parameters, canonical principal parameters, parametric polynomial surface

2010 Mathematics Subject Classification: 53A10