We know that the inflection points of the plane complex curve are exactly the non-singular points where the Hessian determinant is zero. But sometimes Hesse curve coincides with the original curve. The possible example is the curve $x^3+y^3+z^3-3xyz=0$, which is the product of three lines.
Is it true that a curve can be a subset of a Hasse curve only in similar degenerate cases?