The ranks of planarity for varieties of commutative semigroups

We study the concept of the planarity rank suggested by L. M. Martynov for semigroup varieties. Let $V$ be a variety of semigroups. If there is a natural number $r\geq 1$ that all $V$-free semigroups of ranks $\leq r$ allow planar Cayley graphs and the $V$-free semigroup of a rank $r+1$ doesn't allow planar Cayley graph, then this number $r$ is called the planarity rank for variety $V$. If such a number $r$ doesn't exist, then we say that the variety $V$ has the infinite planarity rank. We prove that a non-trivial variety of commutative semigroups either has the infinite planarity rank and coincides with the variety of semigroups with the zero multiplication or has a planarity rank $1$, $2$ or $3$. These estimates of planarity ranks for varieties of commutative semigroups are achievable.