Degree 3 algebraic minimal surfaces in the 3-sphere

We give a local analytic characterization that a minimal surface in the 3-sphere ${\text{ES}}^3\subset {\mathbb{R}}^4$ defined by an irreducible cubic polynomial is one of the Lawson’s minimal tori. This provides an alternative proof of the result by Perdomo (\emph{Characterization of order 3 algebraic immersed minimal surfaces of $S^3$},Geom. Dedicata 129 (2007), 23–34).