Let $H_9\big((x,y),(u,w)\big)$ be a $9\times 9$ complex Hadamard matrix of the following symmetric and block circulant, with circulant blocks ($BCCB$) form
$$H_9\big((x,y),(u,w)\big)=\left[\begin{array}{lll|lll|lll} 1& x& x& y& u& w& y& w& u\\ x& 1& x& w& y& u& u& y& w\\ x& x& 1& u& w& y& w& u& y\\ \hline y& w& u& 1& x& x& y& u& w\\ u& y& w& x& 1& x& w& y& u\\ w& u& y& x& x& 1& u& w& y\\ \hline y& u& w& y& w& u& 1& x& x\\ w& y& u& u& y& w& x& 1& x\\ u& w& y& w& u& y& x& x& 1 \end{array}\right].$$A two-parameter, non-affine family of complex Hadamard matrices found by B. Karlsson [161] stemming from $H_9\big((x,y),(u,w)\big)$ is denoted by $K_9^{(2)}(\zeta)$, where
$$\zeta \in \big\{z\in\mathbb{C} : |1-z|\leqslant 4\big\}\cap\big\{z\in\mathbb{C} : |1+z| \leqslant 4\big\}\setminus\{\mp 1\}$$with
$$ \left.\begin{array}{cc}x\\y\end{array}\right\} = \frac{1}{4}(1+\zeta)\left(1\pm i\sqrt{\frac{16}{|1+\zeta|^2}-1}\right) $$ and $$ \left.\begin{array}{cc}u\\w\end{array}\right\} = \frac{1}{4}(1-\zeta)\left(1\pm i\sqrt{\frac{16}{|1-\zeta|^2}-1}\right). $$In particular, the property of being a complex Hadamard matrix is invariant under the operation of swapping of $x \leftrightarrow y$ and/or $u \leftrightarrow w$.
The defect of a generic element of $K_9^{(2)}$ is $2$, so it is not contained in any other orbit, however it contains two affine $1$-parameter sub-orbits [161].