Degenerate multidimensional dispersion laws

A study is made of the degeneracy of multidimensional dispersion laws $\omega ({\mathbf k})$, increasing infinitely at $|{\mathbf k}|\to \infty$ and satisfying a number of additional conditions is investigated. With the assumption of satisfying condition (4) by corresponding function of degeneracy $f(\mathbf {k})$ it is proved that only two-dimensional dispersion laws such as $\omega (p, q)=p^3\Omega (q/p)+cp\Omega _1(q/p)$ $\bigl (|p|, |q|\gg 1\bigr )$ can be generated relatively to the process $1\to 2$. Here $p\psi (q/p)=f(p, q)$ is the corresponding unique function of degeneracy. Number of conditions were found which should be satisfied by function $\Omega (\xi )$. An explicit form of the degenerate dispersion law with the polynomial function $p^3\Omega (q/p)$ is found.