that $x,y,z$ are likely to share many common factors. To eliminate these common factors, it is convenient to lift $S_n$ to higher-dimensional varieties $\Sigma^{{\operatorname{I}}}_n$, $\Sigma^{{{\operatorname{II}}}}_n$ (and more specifically, to three-dimensional varieties in ${\mathbb{C}}^6$), which are adapted to parameterising Type I and Type II solutions respectively. This will replace the three original coordinates $x,y,z$ by six coordinates $a,b,c,d,e,f$, any three of which can be used to parameterise $\Sigma^I_n$. or $\Sigma^{{{\operatorname{II}}}}_n$. This multiplicity of parameterisations will be useful for many of the applications in this paper; rather than pick one parameterisation in advance, it is convenient to be able to pick and choose between them, depending on the situation.

We begin with the description of Type I solutions. More precisely, we define $\Sigma^{{\operatorname{I}}}_n$ to be the set of all sextuples $(a,b,c,d,e,f) \in {\mathbb{C}}^6$ which are non-zero and obey the constraints $$\begin{aligned}
4abd &= ne+1 \label{I-1}\\
ce &= a+b \label{I-2}\\
4abcd &= na + nb + c \label{I-3}\\
4acde &= ne + 4a^2 d + 1 \label{I-4}\\
4bcde &= ne + 4b^2 d + 1\label{I-5}\\
4acd &= n + f \label{I-6}\\
ef &= 4a^2 d + 1 \label{I-7}\\
bf &= na + c \label{I-8}\\
n^2 + 4c^2d &= f(4bcd-n).\label{I-9}\end{aligned}$$

There are multiple redundancies 