More queens problems (2)

In [1], I discussed the Minimum Dominating Set of Queens Problem. Here is a variant:

Given an $n \times n$ chess board, place $n$ queens on the board such that the number of squares not under attack is maximized. 

For an $n=8$ problem, the maximum number of non-dominated squares is 11. An example is:

11 white pawns are not under attack

We base the model on the one in [1]. First we define two sets of binary variables:

$$\begin{align} & x_{i,j} = \begin{cases} 1 & \text{if square \((i,j)\) is occupied by a queen}\\ 0 & \text{otherwise}\end{cases} \\ & y_{i,j} = \begin{cases} 1 & \text{if square \((i,j)\) is under attack} \\ 0 & \text{otherwise}\end{cases}\end{align}$$

The model can look like:

Maximum Non-Dominated Squares Problem

$$\begin{align}\max& \>\color{DarkRed}z=\sum_{i,j} \left(1-\color{DarkRed} y_{i,j}\right)\\ & \sum_{j'} \color{DarkRed}x_{i,j'} + \sum_{i'} \color{DarkRed}x_{i',j} + \sum_{i',j'|i'-j'=i-j} \color{DarkRed}x_{i',j'} +\sum_{i',j'|i'+j'=i+j} \color{DarkRed}x_{i',j'}-3 \color{DarkRed} x_{i,j} \ge \color{DarkBlue}M \cdot \color{DarkRed}y_{i,j} && \forall i,j \\  & \sum_{i,j}  \color{DarkRed} x_{i,j} = n \\& \color{DarkRed}x_{i,j},\color{DarkRed}y_{i,j}\in\{0,1\} \end{align}$$

The funny term $-3x_{i,j}$ is to compensate for double counting occurrences of $x_{i,j}$ in the previous terms [1]. We need to find a reasonable value for $M$. An upper bound is $M=n$. For $n=8$, there are 48 different optimal solutions.

The problem $n=9$ has one structural optimal solution (with 18 unattacked positions), and 3 variants by symmetry:

Is there a good way to find only structural solutions? One way is to use your own cuts as in [4], If we want to use the solution pool technology, we need to invent a constraint that forbids some symmetry (or have an objective that favors some of symmetric versions).

References

1. More queens problems, https://yetanothermathprogrammingconsultant.blogspot.com/2018/12/more-queens-problems.html
2. Bernard Lemaire, Pavel Vitushinkiy, Placing $n$ non-dominating queens on the $n\times n$ Chessboard, 2011, https://www.ffjm.org/upload/fichiers/N_NON_DOMINATING_QUEENS.pdf
3. Maximum Queens Chess Problem, https://www.gams.com/latest/gamslib_ml/queens.103