For your sample problem, I get two flavors of identity, principal and non-principal.
For discriminant 229, I take the identity form as $$ f(x,y) = x^2 + 15 x y - y^2.$$ One of your families is $$ f(x^3 + 3 x y^2 + 5 y^3, \; 3 x^2 y + 45 x y^2 + 226 y^3 ) \; = \; f^3(x,y).$$ As an automorph of $f$ is $$ \left( \begin{array}{rr} 1 & 15 \\\ 15 & 226 \end{array} \right) , $$ from the column vector $(1,0)^T$ we get another representation of 1 as $(1,15)^T.$ So that is one type of thing.
For the other two classes, take
$$ g(x,y) = 3 x^2 + 13 x y - 5 y^2.$$
A second family is
$$ f( x^3 + 12 x^2 y + 57 x y^2 + 89 y^3, \; 2 x^3 - 3 x^2 y - 3 x y^2 - 6 y^3 ) \; = \; g^3(x,y).$$
$$ $$
$$ $$ The following cycles of reduced forms are as in Buell's book, pages 21-30.
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
1 15 -1
0 form 1 15 -1 delta -15
1 form -1 15 1 delta 15
2 form 1 15 -1
minimum was 1rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 229
Automorph, written on right of Gram matrix:
-1 -15
-15 -226
Trace: -227 gcd(a21, a22 - a11, a12) : 15
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jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2
3 13 -5
0 form 3 13 -5 delta -2
1 form -5 7 9 delta 1
2 form 9 11 -3 delta -4
3 form -3 13 5 delta 2
4 form 5 7 -9 delta -1
5 form -9 11 3 delta 4
6 form 3 13 -5
minimum was 3rep 1 0 disc 229 dSqrt 15.13274595 M_Ratio 25.44444
Automorph, written on right of Gram matrix:
-16 -75
-45 -211
Trace: -227 gcd(a21, a22 - a11, a12) : 15
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