Butson type matrix found by P. Lampio et al. [164]
$${\rm LOG}\left(L_{14}\right)=\frac{1}{5}\pi \left[\begin{array}{rrrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & 8 & 6 & 5 & 4 & 2 & \bullet & \bullet & 8 & 6 & 5 & 4 & 2\\ \bullet & \bullet & 8 & 6 & 5 & 4 & 2 & 5 & 5 & 3 & 1 & \bullet & 9 & 7\\ \bullet & 1 & 3 & 5 & 2 & 9 & 5 & 7 & 9 & 3 & 7 & 7 & 1 & 5\\ \bullet & 2 & 4 & 4 & 9 & 6 & 9 & 9 & 4 & 5 & 1 & 6 & 7 & 1\\ \bullet & 2 & 6 & 2 & 5 & 8 & 7 & 3 & 7 & 1 & 5 & \bullet & 5 & \bullet\\ \bullet & 4 & \bullet & 2 & 3 & 4 & 7 & 9 & 5 & 8 & 7 & 2 & 9 & 5\\ \bullet & 4 & 4 & 8 & 1 & 2 & 3 & 5 & 7 & 9 & 9 & 4 & 6 & 9\\ \bullet & 5 & 8 & \bullet & 1 & 6 & \bullet & 4 & 2 & 4 & 6 & 6 & 2 & 8\\ \bullet & 6 & 2 & 6 & 7 & 2 & 7 & 5 & 3 & 9 & 4 & 8 & 1 & 1\\ \bullet & 6 & 2 & 1 & 7 & 2 & 2 & \bullet & 8 & 4 & 4 & 8 & 6 & 6\\ \bullet & 6 & 7 & 4 & 9 & 8 & 4 & 8 & 2 & \bullet & 2 & 2 & 4 & 6\\ \bullet & 8 & 2 & 8 & 3 & 7 & 8 & 4 & \bullet & 6 & 2 & 2 & 6 & 4\\ \bullet & 8 & 6 & \bullet & 7 & 0 & 5 & 2 & 5 & 5 & 9 & 4 & 1 & 3\\ \end{array}\right].$$Define
$$R_{L_{14}^{(2)}}(a,b)=\left[\begin{array}{rrrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & b & b & b & b & b & b & b\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & b & b & b & b & b & b & b\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & a & \bullet & \bullet & a & a & a & a & \bullet & \bullet & a & a\\ \bullet & \bullet & \bullet & a & \bullet & \bullet & a & a & a & a & \bullet & \bullet & a & a\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \end{array}\right]$$so that, for $a,b\in[0,1)$ one has $2$-parametric affine family.
$$L_{14}\mapsto L_{14}^{(2)}(a,b) = L_{14} \circ {\rm EXP}\left(2 \pi i R_{L_{14}^{(2)}}(a,b)\right).$$$L_{14}$ has two ER pairs of rows [112] and belongs to a $2$-dimensional affine family. The defect of $L_{14}$ is $10$ so this orbit might be embedded in a higher-dimensional (probably non-affine) structure.