Quasi-periodic solutions of a nonlinear periodic polyharmonic equation in
$\R^n$, $n>1$, are studied. It is proven that there is an extensive
"non-resonant" set ${\mathcal G}\subset \R^n$ such that for every $\vec k\in
\mathcal G$ there is a solution asymptotically close to a plane wave
$Ae^{i\langle{ \vec{k}, \vec{x} }\rangle}$ as $|\vec k|\to \infty $.