equation $$\label{CMeq}X^{-1}Y^{-1}XY\tau-\tau^{-1}=U\otimes V.$$ Obviously it is an affine variety.

The group $GL(n,\mathbb C)=GL(E)$ acts on it by conjugation: $$(X,Y,U,V)\to (gXg^{-1},gYg^{-1},gU,Vg^{-1}),\quad g\in GL(E).$$ Later we will show that this action is free if $\tau$ is not a root of unity. So the naive quotient by the action (i.e. the spectrum of the ring of $GL(E)$ invariant functions) yields an affine variety, and the quotient is nonsingular if $CM'_\tau$ is.

The quotient of $CM'_\tau$ by the action $GL(E)$ is called the Calogero-Moser space. We use the notation $CM_\tau$ for this space.

Below we always suppose that $\tau$ is not a root of unity.

Properties of the Calogero-Moser space
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The goal of this section is to prove that $CM_\tau$ is a smooth irreducible algebraic variety of dimension $2n$. We also introduce coordinates on its dense subset. The methods of this section are analogous to the ones from the paper [@W]. In principal smoothness of $CM_\tau$ follows from the results of the paper [@FR], the authors of [@FR] use the moduli space of the vector bundles on the punctured torus. For convenience of reader we give a direct elementary proof.

Smoothness of the Calogero-Moser space
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First we prove a simple lemma on 