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We construct new families of two-ended $O(m)\times O(n)$-invariant solutionsto the Allen- Cahn equation \Delta u+u-u3=0 in $\mathbb{R}^{N+1}$, with $N\ge7$, whose zero level sets diverge logarithmically from the Lawson cone atinfinity. The construction is based on a careful study of the Jacobi-Todasystem on a given $O(m)\times O(n)$-invariant manifold, which is asymptotic tothe Lawson cone at infinity.

Doubling construction for $O(m)\times O(n)$-invariant solutions to the Allen-Cahn equation