  We describe the $\nu$-lines of curvature of an embedding of the double torus
into $\mathbb R^4$, defined as the link of the real part of the Milnor
fibration of a polynomial, where $\nu$ is its gradient. Through this analysis,
we present a complete description of the foliation of lines of curvature of the
embedding, defined as the image of the stereographic projection of this link
into $\mathbb R^3$.
