  In this work we study existence, asymptotic behaviour and stability
properties of $O(m) times O(n)$ invariant solutions of the Allen-Cahn equation
$Delta u+u(1-u^2)=0$ in $R^m times R^n$ with $m,n ge 2$, $m+n ge 8$. We exhibit
four families of solutions whose nodal sets are smooth logarithmic corrections
of the Lawson cone and with infinite Morse index. This work complements the
study started by Pacard and Wei (about stable solutions) and by Agudelo,
Kowalczyk and Rizzi (about 2 ended solutions).
