\delta x_i & = & \sum_{\alpha=1}^6 \mathring r_{i\alpha} e^{\imath \theta_\alpha} \\
  \delta p_i & = & \sum_{\alpha=1}^6 \hat r_{i\alpha} e^{\imath \theta_\alpha}\end{aligned}$$ with $\mathring r_{i\alpha} = c_\alpha \mathring e_{\alpha i}$ if $\alpha \leq 3$, $\mathring r_{i\alpha} = d_{\alpha-3} \mathring e_{\alpha-3, i}^*$ if $\alpha > 3$. It follows that $$\delta \dot p_i = \imath \sum_{\alpha=1}^6 \hat r_{i\alpha} e^{\imath \theta_\alpha} \dot \theta_\alpha$$ Finally we write $\delta \dot p_i = \imath \hat R \dot \theta$ where the matrix $\hat R$ is defined by the elements $\hat r_{i\alpha} e^{\imath \theta_\alpha}$. We have then $\dot \theta = - \imath \hat R^{-1} \delta \dot p$. The dynamics onto $\mathbb T^6$ is then defined by the equation $$\dot \theta = {\Im\mathrm{m}}\hat R(\theta)^{-1} \hat F(\chi(\theta))$$ where $\chi : \mathbb T^6 \to \mathbb C^3$ is defined by $\chi(\theta) = \sum_{\alpha=1}^3 (c_\alpha e^{\imath \theta_\alpha} \mathring e_\alpha + d_\alpha e^{\imath \theta_{\alpha+3}} \mathring e_\alpha^*)$, and $\hat F:\mathbb C^3 \to \mathbb C^3$ is defined by $\hat F_i(\delta x) = - \tr |X-x|^2 \delta x_i + \tr \left((X^j-x^j)(X_i-x_i)\right)\delta x_j - 2 \overline{\delta x^j} \delta x_j \delta x_i + 2 \overline{\delta x_i} \delta x^j \delta x_j$.\
Finally we have $\dot \theta = F(\theta)$ with $$\label{eqF}
  F = {\Im\mathrm{m}}\hat R^{-1} \hat F \circ \chi$$

Remark: note that 