subject (dating back as far as 1994). Both authors are particularly indebted to him for several remarks (including Remark \[heath-remark\]), and also for contributing some of the key arguments here (such as the lower bound on $\sum_{n {\leqslant}N} f_{{\operatorname{II}}}(n)$ and $\sum_{p {\leqslant}N}
f_{{\operatorname{II}}}(p)$) which have been reproduced here with permission. The first author also wishes to thank Tim Browning, Ernie Croot and Arnd Roth for discussions on the subject.

Representation of Type I and Type II solutions {#represent-sec}
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We now discuss the representation of Type I and Type II solutions. There are many such representations in the literature (see e.g. [@aigner], [@bello], [@bernstein], [@nakayama], [@rav], [@rosati], [@vaughan], [@webb]); we will remark how each of these representations can be viewed as a form of the one given here after describing a certain algebraic variety in coordinates.

For any non-zero complex number $n$, consider the algebraic surface $$S_n := \{ (x,y,z) \in {\mathbb{C}}^3: 4xyz = nyz + nxz + nxy\} \subset {\mathbb{C}}^3.$$ Of course, when $n$ is a natural number, $f(n)$ is nothing more than the number of ${\mathbb{N}}$-points $(x,y,z) \in S_n \cap {\mathbb{N}}^3$ on this surface.

It is somewhat inconvenient to count ${\mathbb{N}}$-points on $S_n$ directly, due to the fact 