The structure of locally finite varieties with polynomially many models

We prove that a locally finite variety has at most polynomially many (in $k$ ) non-isomorphic $k$ –generated algebras if and only if it decomposes into a varietal product of an affine variety over a ring of finite representation type, and a sequence of strongly Abelian varieties equivalent to matrix powers of varieties of $H$ -sets, with constants, for various finite groups $H$ .