A quadratic $O(N)$ invariant equation for 4-index tensors Consider an $O(N)$ invariant quadratic equation
$$
T_{ijkl}= T_{ijmn}T_{klmn}+ T_{ikmn}T_{jlmn}+ T_{ilmn}T_{jkmn},
$$
where $T_{ijkl}$ is a real, totally symmetric 4-tensor, and the indices run from 1 to $N$.
[Although this is not important for what follows, this equation appears in theoretical physics where it describes renormalization group fixed points in quantum field theory of $N$ scalar fields in $d=4-\epsilon$ dimensions, in the lowest order of perturbative expansion in the interaction strength.]
I have two questions about this equation, one computational for small $N$, and one more structural which involves arbitrarily large $N$.

*

*Is there an efficient algorithm to fully classify the set of solutions (up to $O(N)$ transformations), which one could apply for small $N$? Physicists have achieved the full classification only up $N=3$, by brute force methods which seem hard to extend to larger $N$.


*For any solution $T$ of the above equation, it is interesting to ask what is its symmetry, i.e. what is its invariance subgroup $G\subset O(N)$. Any even rank tensor has invariance subgroup at least as large as $\mathbb{Z}_2$, generated by the inversion transformation $x\to -x$. Many distinct solutions of the equation, for various $N$, are known in the physics literature. By inspection, all of them have an invariance subgroup $G\subset O(N)$ (some continuous, some discrete) which is strictly larger than $\mathbb{Z}_2$. Is there a deep reason behind this fact? If there exists a solution, for some $N$, with the invariance subgroup just $\mathbb{Z}_2$, how can one find it?
 A: Perhaps I might add a few more comments related to my work with Andy Stergiou, in particular that related to archive 2010.15915 where we
discussed in some detail various examples for N up to 7.
For N=4 an analysis based on an O(4) decomposition was previously
carried by Codello et al in arXiv:2008.04077 and they found at least two hitherto unknown irrational solutions. There is a third which is of
the biconical type. Numerical analysis suggests all solutions are known
when N=4 and perhaps for 5,6. From our results for N=4 there are 15 reducible solutions and 5 irreducible ones, 3 are irrational.
In general the tensor T can be decomposed into symmetric traceless 4 and 2 index tensors, as in T4 and T2, as well as a scalar. If the 2 index tensor is zero things are much simpler. However in this case for N  non prime there are
solutions which can be obtained from the symmetric product of lower order solutions and perturbing them by sums of products of all possible   invariant 2 index  tensors from the different factors (not very well explained but like the wreath product for groups). Even for N=6 this gives quite a few. For N prime it might be possible to classify solutions. At present the known solutions are based on hypercubic or hypertetrahedral symmetry in addition to the fully symmetric O(N) solution.
With a non zero 2 index tensor present things are much more involved and typically this is where there are irrational solutions in general.
I Could say quite a bit more. A very similar problem based on 3 index tensors is discussed by Cvitanovic in his book. I this case there are just 4 solutions which appear to be related to division algebras.
