The variety $\mathbf N_3\mathbf N_2$ of commutative alternative nil-algebras of index 3 over a field of characteristic $3$

A variety is called a Specht variety if every algebra in this variety has a finite basis of identities. In 1981 S. V. Pchelintsev defined the topological rank of a Specht variety. Let $\mathbf N_k$ be the variety of commutative alternative algebras over a field of characteristic 3 with nilpotency class not greater than $k$. Let $\mathbf D$ be the variety $\mathbf N_3\mathbf N_2$ of nil-algebras of index 3, i.e. the commutative alternative algebras with identities
$$ x^3=0,\quad [(x_1x_2)(x_3x_4)](x_5x_6)=0. $$
In the paper we prove that the varieties $\mathbf N_k\mathbf N_l$ are Specht varieties. Moreover, a base of the space of polylinear polynomials in the free algebra $F(\mathbf D)$ is built and the topological rank $\mathrm r_{\mathrm t}(\mathbf D_n)=n+2$ of varieties
$$ \mathbf D_n=\mathbf D\cap\mathrm{Var}((xy\cdot zt)x_1\ldots x_n) $$
is found. This implies that the topological rank of the variety $\mathbf D$ is infinite.