There exists a continuous $2$-parameter orbit $$P_{13}^{(2)}(e,f)=P_{13}\circ{\rm EXP}\left(i R_{P_{13}^{(2)}}(e,f)\right)$$ found by M. Petrescu [82], where $$P_{13}=\left[\begin{array}{rrrrrrrrrrrrr} 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1& 1\\ 1& -1& t^{10}& -t^5& t^5& i t^5& -i t^5& i t^{15}& -i t^{15}& t^{16}& t^4& t^{22}& t^{28}\\ 1& t^{10}& -1& t^5& -t^5& -i t^5& i t^5& -i t^{15}& i t^{15}& t^{16}& t^4& t^{22}& t^{28}\\ 1& -t^5& t^5& -1& t^{10}& i t^{15}& -i t^{15}& i t^5& -i t^5& t^4& t^{16}& t^{28}& t^{22}\\ 1& t^5& -t^5& t^{10}& -1& -i t^{15}& i t^{15}& -i t^5& i t^5& t^4& t^{16}& t^{28}& t^{22}\\ 1& i t^5& -i t^5& i t^{25}& -i t^{25}& -1& t^{10}& -t^5& t^5& t^{22}& t^{28}& t^4& t^{16}\\ 1& -i t^5& i t^5& -i t^{25}& i t^{25}& t^{10}& -1& t^5& -t^5& t^{22}& t^{28}& t^4& t^{16}\\ 1& i t^{25}& -i t^{25}& i t^5& -i t^5& -t^5& t^5& -1& t^{10}& t^{28}& t^{22}& t^{16}& t^4\\ 1& -i t^{25}& i t^{25}& -i t^5& i t^5& t^5& -t^5& t^{10}& -1& t^{28}& t^{22}& t^{16}& t^4\\ 1& t^4& t^4& t^{16}& t^{16}& t^{28}& t^{28}& t^{22}& t^{22}& t^{20}& t^{10}& t^{10}& t^{10}\\ 1& t^{16}& t^{16}& t^4& t^4& t^{22}& t^{22}& t^{28}& t^{28}& t^{10}& t^{20}& t^{10}& t^{10}\\ 1& t^{28}& t^{28}& t^{22}& t^{22}& t^{16}& t^{16}& t^4& t^4& t^{10}& t^{10}& t^{20}& t^{10}\\ 1& t^{22}& t^{22}& t^{28}& t^{28}& t^4& t^4& t^{16}& t^{16}& t^{10}& t^{10}& t^{10}& t^{20} \end{array}\right] : \quad t=\exp\frac{2i\pi}{30}$$ and $$R_{P_{13}^{(2)}}(e,f)=\left[\begin{array}{rrrrrrrrrrrrr} \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & f& f& e& e& e + G(f)& e + G(f)& \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & f& f& e& e& e + G(f)& e + G(f)& \bullet & \bullet & \bullet & \bullet\\ \bullet & f& f& \bullet & \bullet & e + G(f)& e + G(f)& e& e& \bullet & \bullet & \bullet & \bullet\\ \bullet & f& f& \bullet & \bullet & e + G(f)& e + G(f)& e& e& \bullet & \bullet & \bullet & \bullet\\ \bullet & -e& -e& -e + G(f)& -e + G(f)& \bullet & \bullet & -f& -f& \bullet & \bullet & \bullet & \bullet\\ \bullet & -e& -e& -e + G(f)& -e + G(f)& \bullet & \bullet & -f& -f& \bullet & \bullet & \bullet & \bullet\\ \bullet & -e + G(f)& -e + G(f)& -e& -e& -f& -f& \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & -e + G(f)& -e + G(f)& -e& -e& -f& -f& \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet\\ \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet & \bullet \end{array}\right],$$ where $$G(f)=\arg\left(-\frac{\cos(f)}{2}+i\frac{\sqrt{2}}{4}\sqrt{7-\cos(2f)}-\frac{2}{3}\pi\right).$$ Since the function $G(f)$ is nonlinear, the above family is not an affine Hadamard family.
update 2011/07/16
F. Szöllősi [37] has extended it to the following $4$-parameter family: $$ \begin{aligned} P_{13A}^{(4)}(a,b,c,d) & =\\ & \left[\begin{array}{cccc|cccc|ccccc} \omega & a & b & c & -1 & -a & -b & -c & w & w^2 & w^3 & w^4 & 1\\ \frac{d^2\omega}{a} & \omega & -\frac{bde}{a} & -\frac{cd\omega}{ae} & -\frac{d^2\omega}{a} & -1 & \frac{bde}{a} & \frac{cd\omega}{ae} & w^2 & w^4 & w & w^3 & 1\\ \frac{\omega}{b} & -\frac{a\omega}{bde} & -1 & \frac{c\omega}{bde} & -\frac{\omega}{b} & \frac{a\omega}{bde} & \omega & -\frac{c\omega}{bde} & w^3 & w & w^4 & w^2 & 1\\ \frac{\omega}{c} & -\frac{ae}{cd} & \frac{be}{cd} & -1 & -\frac{\omega}{c} & \frac{ae}{cd} & -\frac{be}{cd} & \omega & w^4 & w^3 & w^2 & w & 1\\ \hline -1 & -a & -b & -c & \omega & a & b & c & w & w^2 & w^3 & w^4 & 1\\ -\frac{d^2\omega}{a} & -1 & \frac{bde}{a} & \frac{cd\omega}{ae} & \frac{d^2\omega}{a} & \omega & -\frac{bde}{a} & -\frac{cd\omega}{ae} & w^2 & w^4 & w & w^3 & 1\\ -\frac{\omega}{b} & \frac{a\omega}{bde} & \omega & -\frac{c\omega}{bde} & \frac{\omega}{b} & -\frac{a\omega}{bde} & -1 & \frac{c\omega}{bde} & w^3 & w & w^4 & w^2 & 1\\ -\frac{\omega}{c} & \frac{ae}{cd} & -\frac{be}{cd} & \omega & \frac{\omega}{c} & -\frac{ae}{cd} & \frac{be}{cd} & -1 & w^4 & w^3 & w^2 & w & 1\\ \hline w^4 & w^3 & w^2 & w & w^4 & w^3 & w^2 & w & 1 & \omega^2 & \omega^2 & \omega^2 & \omega^2\\ w^3 & w & w^4 & w^2 & w^3 & w & w^4 & w^2 & \omega^2 & 1 & \omega^2 & \omega^2 & \omega^2\\ w^2 & w^4 & w & w^3 & w^2 & w^4 & w & w^3 & \omega^2 & \omega^2 & 1 & \omega^2 & \omega^2\\ w & w^2 & w^3 & w^4 & w & w^2 & w^3 & w^4 & \omega^2 & \omega^2 & \omega^2 & 1 & \omega^2\\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & \omega^2 & \omega^2 & \omega^2 & \omega^2 & 1 \end{array}\right], \end{aligned}$$ where $w=\exp(2\pi i/5)$, $\omega=\exp(2\pi i/3)$ and ${\rm Re}(e\omega)={\rm Re}(d)/2.$ Watch the different symbols in use: $w$ and $\omega$ (omega!).