Doubling nodal solutions to the Yamabe equation in ${\mathbb{R}}^n$ with maximal rank

We construct a new family of entire solutions to the Yamabe equation $$-\mathrm{\Delta }u=\frac{n(n-2)}{4}\mathrm{\mid }u{\mathrm{\mid }}^{\frac{4}{n-2}}uin{\mathcal{D}}^{1,2}({\mathbb{R}}^n)\mathrm{.}$$ If $n=3$, our solutions have maximal rank, being the first example in odd dimension. Our construction has analogies with the doubling of the equatorial spheres in the construction of minimal surfaces in $S^3(1)$.