A non-affine two-parameter family of complex Hadamard matrices found by B. Karlsson [105] reads $$K_6^{(2)}(x_1,x_2)=\left[\begin{array}{rrrrrr} 1 & 1 & 1 & 1 & 1 & 1\\ 1 & -1 & z_1 & -z_1 & z_1 & -z_1\\ 1 & z_2 & -f_1 & -z_2f_2 & -\bar{f_3} & -z_2\bar{f_4}\\ 1 & -z_2 & -z_1\bar{f_2} & z_1z_2\bar{f_1} & -z_1f_4 & z_1z_2f_3\\ 1 & z_2 & -\bar{f_3} & -z_2\bar{f_4} & -f_1 & -z_2f_2\\ 1 & -z_2 & -z_1f_4 & z_1z_2f_3 & -z_1\bar{f_2} & z_1z_2\bar{f_1} \end{array}\right]$$ with parameters: $z_k=\exp(ix_k): -\pi/2<x_k\leqslant\pi/2$. The elements are given in terms of four factors: $f_1 = f( x_1, x_2)$, $f_2 = f( x_1,-x_2)$, $f_3 = f(-x_1,-x_2)$, $f_4 = f(-x_1, x_2)$, where $$f(x_1,x_2)=\exp\left(i\frac{1}{2}(x_1+x_2)\right) \left(\cos\frac{x_1-x_2}{2}-i\sin\frac{x_1+x_2}{2}\right) \left(\frac{1}{2}+i\sqrt{\frac{1}{1+\sin x_1\sin x_2}-\frac{1}{4}}\right).$$
This family contains a subfamily $K_6(x, 0)$ in common with the Fourier family, another subfamily $K(0, x)$ in common with the Fourier-transposed family, a symmetric subfamily $K(x, x)$ that coincides with $M_6^{(1)}$, and it also contains the Diţă's family $D_6^{(1)}$ at the points $x_1=\pm\pi/2$ and $x_2=\pm\pi/2$.