  An algebraic isopair is a commuting pair of pure isometries that is
annihilated by a polynomial defining a distinguished variety $\mathcal{V}$. The
notion of the rank of a pure algebraic isopair with finite bimultiplicity is
introduced. For $\mathcal{V} $, a union of $s$ irreducible varieties
$\mathcal{V}_j$, the rank is a $s$-tuple $\alpha=(\alpha_1,...,\alpha_s)$ of
natural numbers. A pure algebraic isopair of finite bimultiplicity with rank
$\alpha$ is described as a restriction of a
$\max\{\alpha_1,...,\alpha_s\}$-cyclic pure algebraic isopair to a finite
codimensional invariant subspace. The restriction of a pure algebraic isopair
of finite bimultiplicity with rank $\alpha$ to a finite codimensional invariant
subspace is at least $\max\{\alpha_1,...,\alpha_s\}$-cyclic and there is a
$\max\{\alpha_1,...,\alpha_s\}$-cyclic finite codimensional invariant subspace.
