Thank you for your excellent answer. How would one go about finding a solution to $M_x = -MU$? I originally hoped to write $X(x) = Ce^{-\int M(x) dx}$ but was later told that the lack of commutativity for matrix multiplication causes the chain rule to break down. Would I have to use a Magnus series or consider the system as a time-ordered exponential? Thank you again for all of your help.

Yes, please! I would appreciate any help regarding the problem. Yesterday, I found the following article which provides a parallel development of the problem on page 4. The author states that it is possible but does not provide the particulars of proving it. Thank you again. iopscience.iop.org/article/10.1088/0951-7715/18/4/019

Thank you, I fixed the capitalization issue. I'm sorry, I don't follow what you mean with "$|q|^2$ in $V$ is missing a factor of $\frac{1}{2}$". Do you want me to define $V(x)=2kQ-iQ_{x}\sigma_3 - i \frac{1}{2}|q|^2 \sigma_3 - 2ik^2 \sigma_3$? Thanks again