Consider the following equation system:
$\boldsymbol{x}_i = \boldsymbol{KPT_jTX_i}$
where $\boldsymbol{x}_i$ is a $(3 \times 1)$ vector, $\boldsymbol{K}$ is a $(3 \times 3)$ matrix, $\boldsymbol{P}$ is a $(3 \times 4)$ matrix, $\boldsymbol{T_j}$ and $\boldsymbol{T}$ are $(4 \times 4)$ matrices, $\boldsymbol{X}_i$ is a $(4 \times 1)$ vector. Furthermore, $i \in [1, 6]$ and $j \in [1, 3]$, such that $i=1, i=2$ correspond to $j=1$, $i=3, i=4$ correspond to $j=2$ etc. $\boldsymbol{T}$ only has 6 degrees of freedom and is of the form
$T = \left(\begin{matrix} \boldsymbol{R} & \boldsymbol{t} \\\boldsymbol{0} & 1 \end{matrix}\right)$
where $\boldsymbol{R}$ is a $(3 \times 3)$ rotation matrix ($3$ dof), $\boldsymbol{t} = (t_1, t_2, t_3)^T$, $\boldsymbol{0} = (0, 0, 0)$.
$\boldsymbol{x}_i, \boldsymbol{K}, \boldsymbol{P}, \boldsymbol{T}_j$ and $\boldsymbol{X}_i$ are given $\forall i, j$. How can one solve these six given equations for $\boldsymbol{T}$? I have solved similar equation systems before by vectorizing the matrix and stacking the given equations on top of each other to form the system matrix, but they had different structure from this one and I am really out of ideas here.