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 $��=(��_1,...,��_s)$ of natural numbers. A pure algebraic isopair of finite bimultiplicity with rank $��$ is described as a restriction of a $\max\{��_1,...,��_s\}$-cyclic pure algebraic isopair to a finite codimensional invariant subspace. The restriction of a pure algebraic isopair of finite bimultiplicity with rank $��$ to a finite codimensional invariant subspace is at least $\max\{��_1,...,��_s\}$-cyclic and there is a $\max\{��_1,...,��_s\}$-cyclic finite codimensional invariant subspace.

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