On a property of homogeneous Gaussian $L$-fields

Basing on a theorem due to Yu. A. Rozanov we show that for $n=1$ and for $n=2$ there are only regular and singular homogeneous Gaussian fields $(\xi_t)_{t\in Z^n}$ but for $n\ge 3$ there exist homogeneous Gaussian $L$-fields $(\xi_t)_{t\in Z^n}$ which are neither regular nor singular.