Thursday: here is an example I proved in full detail, that illustrates the use of the mappings in one direction, along with the possible intricacy of the difference between finding all rational null vectors and successfully finding all primitive integer null vectors:

All solutions to $$ 2(x^2 + y^2 + z^2) - 113 (yz+zx+xy) = 0 $$ with $$ \gcd(x,y,z) = 1 $$ can be written as one of four recipes, with the understanding that we sort by absolute value and possibly multiply through by $-1$ so as to demand $x \geq |y| \geq |z|,$

$$
\left(
\begin{array}{r}
x \\
y \\
z
\end{array}
\right) =
\left(
\begin{array}{r}
37 u^2 + 51 uv + 8 v^2 \\
8 u^2 -35 uv -6 v^2 \\
-6 u^2 + 23 uv + 37 v^2
\end{array}
\right)
$$

$$
\left(
\begin{array}{r}
x \\
y \\
z
\end{array}
\right) =
\left(
\begin{array}{r}
32 u^2 + 61 uv + 18 v^2 \\
18 u^2 -25 uv -11 v^2 \\
-11 u^2 + 3 uv + 32 v^2
\end{array}
\right)
$$

$$
\left(
\begin{array}{r}
x \\
y \\
z
\end{array}
\right) =
\left(
\begin{array}{r}
38 u^2 + 45 uv + 4 v^2 \\
4 u^2 -37 uv -3 v^2 \\
-3 u^2 + 31 uv + 38 v^2
\end{array}
\right)
$$

$$
\left(
\begin{array}{r}
x \\
y \\
z
\end{array}
\right) =
\left(
\begin{array}{r}
29 u^2 + 63 uv + 22 v^2 \\
22 u^2 -19 uv -12 v^2 \\
-12 u^2 -5 uv + 29 v^2
\end{array}
\right)
$$

In all four cases we simply discard occurrences when the resulting $x,y,z$ have a common factor.

The four are all of the form $X = R U,$ where
$$ X =
\left(
\begin{array}{r}
x \\
y \\
z
\end{array}
\right)
$$
and
$$ U =
\left(
\begin{array}{r}
u^2 \\
uv \\
v^2
\end{array}
\right).
$$
Clearly we take $\gcd(u,v) = 1.$ We can also take $u,v \geq 0.$ This is an artifact of the extreme symmetry of the ternary form and the extremely special form of the four matrices $R$ that I chose.

```
jagy@phobeusjunior:~$ ./isotropy_just_ordered 2 113 1200
29 22 -12 % B lambda 0 / B lambda 1 = 1
32 18 -11 % B lambda 0 / B lambda 1 = 1
37 8 -6 % B lambda 0 / B lambda 1 = 1
38 4 -3 % B lambda 0 / B lambda 1 = 1
188 171 -86 % B lambda 0 / B lambda 49 = 7^2
211 144 -82 % B lambda 0 / B lambda 49 = 7^2
226 123 -76 % B lambda 0 / B lambda 49 = 7^2
243 94 -64 % B lambda 0 / B lambda 49 = 7^2
246 88 -61 % B lambda 0 / B lambda 49 = 7^2
258 59 -44 % B lambda 0 / B lambda 49 = 7^2
264 38 -29 % B lambda 0 / B lambda 49 = 7^2
268 11 -6 % B lambda 0 / B lambda 49 = 7^2
396 262 -151 % B lambda 0 / B lambda 169 = 13^2
432 209 -134 % B lambda 0 / B lambda 169 = 13^2
472 129 -94 % B lambda 0 / B lambda 169 = 13^2
489 76 -58 % B lambda 0 / B lambda 169 = 13^2
516 458 -233 % B lambda 0 / B lambda 361 = 19^2
526 447 -232 % B lambda 0 / B lambda 361 = 19^2
628 311 -198 % B lambda 0 / B lambda 361 = 19^2
656 262 -177 % B lambda 0 / B lambda 361 = 19^2
671 232 -162 % B lambda 0 / B lambda 361 = 19^2
692 183 -134 % B lambda 0 / B lambda 361 = 19^2
726 47 -32 % B lambda 0 / B lambda 361 = 19^2
727 36 -22 % B lambda 0 / B lambda 361 = 19^2
804 787 -382 % B lambda 0 / B lambda 961 = 31^2
894 688 -373 % B lambda 0 / B lambda 961 = 31^2
953 946 -456 % B lambda 0 / B lambda 1369 = 37^2
1034 492 -317 % B lambda 0 / B lambda 961 = 31^2
1062 443 -296 % B lambda 0 / B lambda 961 = 31^2
1102 363 -256 % B lambda 0 / B lambda 961 = 31^2
1123 314 -228 % B lambda 0 / B lambda 961 = 31^2
1159 1046 -528 % B lambda 0 / B lambda 1849 = 43^2
1179 118 -88 % B lambda 0 / B lambda 961 = 31^2
1188 19 2 % B lambda 0 / B lambda 961 = 31^2
1199 1002 -524 % B lambda 0 / B lambda 1849 = 43^2
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
```

hi

```
jagy@phobeusjunior:~$ ./isotropy_binaries_combined 2 113 1200 | sort -n
x y z recipe u v
29 22 -12 < 29, 63, 22 > 1 0
32 18 -11 < 32, 61, 18 > 1 0
37 8 -6 < 37, 51, 8 > 1 0
38 4 -3 < 38, 45, 4 > 1 0
188 171 -86 < 37, 51, 8 > 1 2
211 144 -82 < 38, 45, 4 > 1 2
226 123 -76 < 32, 61, 18 > 1 2
243 94 -64 < 29, 63, 22 > 1 2
246 88 -61 < 38, 45, 4 > 2 1
258 59 -44 < 37, 51, 8 > 2 1
264 38 -29 < 29, 63, 22 > 2 1
268 11 -6 < 32, 61, 18 > 2 1
396 262 -151 < 37, 51, 8 > 1 3
432 209 -134 < 38, 45, 4 > 1 3
472 129 -94 < 29, 63, 22 > 3 1
489 76 -58 < 32, 61, 18 > 3 1
516 458 -233 < 38, 45, 4 > 2 3
526 447 -232 < 37, 51, 8 > 2 3
628 311 -198 < 38, 45, 4 > 3 2
656 262 -177 < 32, 61, 18 > 2 3
671 232 -162 < 37, 51, 8 > 3 2
692 183 -134 < 29, 63, 22 > 2 3
726 47 -32 < 32, 61, 18 > 3 2
727 36 -22 < 29, 63, 22 > 3 2
804 787 -382 < 32, 61, 18 > 1 5
894 688 -373 < 29, 63, 22 > 1 5
953 946 -456 < 38, 45, 4 > 3 4
1034 492 -317 < 37, 51, 8 > 1 5
1062 443 -296 < 29, 63, 22 > 5 1
1102 363 -256 < 38, 45, 4 > 1 5
1123 314 -228 < 32, 61, 18 > 5 1
1159 1046 -528 < 32, 61, 18 > 1 6
1179 118 -88 < 38, 45, 4 > 5 1
1188 19 2 < 37, 51, 8 > 5 1
1199 1002 -524 < 29, 63, 22 > 1 6
```

well, then. I put in some blank lines..