Smooth 2-homogeneous polynomials on the plane with a hexagonal norm

Motivated by the classifications of extreme and exposed 2-homogeneous polynomials on the plane with the hexagonal norm ||(x, y)|| = max{|y|, |x| + |y|/2} (see [15, 16]), we classify all smooth 2-homogeneous polynomials on R2 with the hexagonal norm.