Spent a month checking, this is what I suspect is the complete list of 'sporadic' or 'exceptional' pairs. No restriction that they be in the same genus or have the same discriminant. I was able to check discriminant ratio $4$ and discriminant ratio $1$ very high. The other ones just seem sort of random little sets, two quadruples (see the repeated forms). Also some patterns that dry up, discriminants $111,333,999$ but not $2997,$ also on the left $24, 72, 216, 648, 1944,$ but not $5832.$ This last begins with three pair of regular forms, I typed those in. The list of pairs of regular forms that agree is enormous.
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111 : 1 4 7 1 0 0 75 : 1 4 5 1 1 0
142 : 3 3 5 2 3 1 78 : 3 3 3 1 1 3
158 : 3 3 5 -1 2 1 78 : 3 3 3 1 1 3
158 : 3 3 5 -1 2 1 142 : 3 3 5 2 3 1
190 : 3 5 5 5 2 3 78 : 3 3 3 1 1 3
190 : 3 5 5 5 2 3 142 : 3 3 5 2 3 1
190 : 3 5 5 5 2 3 158 : 3 3 5 -1 2 1
213 : 2 4 7 0 1 1 177 : 2 4 7 4 2 1
216 : 2 4 8 4 1 1 54 : 2 2 4 1 2 0
232 : 3 5 5 3 1 3 232 : 3 3 7 1 2 1
284 : 3 5 6 4 2 2 156 : 3 3 5 2 2 0
316 : 3 5 6 0 2 2 156 : 3 3 5 2 2 0
316 : 3 5 6 0 2 2 284 : 3 5 6 4 2 2
333 : 3 4 7 1 0 0 225 : 3 4 7 4 3 3
380 : 3 5 7 2 0 2 156 : 3 3 5 2 2 0
380 : 3 5 7 2 0 2 284 : 3 5 6 4 2 2
380 : 3 5 7 2 0 2 316 : 3 5 6 0 2 2
567 : 4 6 7 3 2 3 324 : 4 4 6 0 3 2
639 : 5 5 8 -1 2 4 531 : 5 5 6 0 3 2
648 : 2 6 14 3 1 0 162 : 2 2 14 1 2 2
648 : 5 7 7 6 1 5 648 : 5 5 8 0 4 3
999 : 5 8 8 -5 1 4 675 : 5 5 8 -1 4 2
1944 : 2 6 41 3 1 0 486 : 2 2 41 1 2 2
2592 : 4 7 25 -4 2 2 648 : 4 7 7 5 2 2
These are pairs of positive quadratic forms that represent the
same numbers, and violate a Kaplansky conjecture.
Delta : A B C R S T means
f(x,y,z) = A x^2 + B y^2 + C z^2 + R y z + S z x + T x y,
and Delta = 4ABC + RST - A R^2 - B S^2 - C T^2.
The two pair within a genus each are
232 : 3 5 5 3 1 3 232 : 3 3 7 1 2 1
648 : 5 7 7 6 1 5 648 : 5 5 8 0 4 3
The most productive discriminant ratio is 4,
which includes Kap's two infinite families, also
24 : 1 2 4 2 1 1 6 : 1 1 2 1 1 0
72 : 2 2 5 1 1 1 18 : 2 2 2 1 2 2
216 : 2 5 6 3 0 1 54 : 2 2 5 1 2 2
648 : 2 6 14 3 1 0 162 : 2 2 14 1 2 2
1944 : 2 6 41 3 1 0 486 : 2 2 41 1 2 2
or
48N-24: 2 6 N 3 1 0 12N-6: 2 2 N 1 2 2
where N = (1+ 3^k)/2, and the pairs for N = 1,2,5 are regular
and have been Schiemann reduced.
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Reminder: Kap's two infinite families are equivalent to those
below, which need not be "reduced." For the first, require
gcd(A,C) = 1 and 0 <A,C. For the second, gcd(A,R) = 1, with
A > 0 and -A < R < 2 A.
4D : A 3A C 0 0 0 D : A A C 0 0 A
4D: A 2A-R 2A+R 0 2R 0 D : A A A R R R
For the first, D = 3 A^2 C, for the second D = (A+R)(2A-R)^2 .