I have an equation with one unknown matrix variable $\mathbf{M}$:
$$ \frac{1}{2}\operatorname{vec}(\mathbf{W})^T(\mathbf{V} \otimes \mathbf{U})\operatorname{vec}(\mathbf{M}) + \frac{1}{2}\operatorname{vec}(\mathbf{M})^T(\mathbf{V} \otimes \mathbf{U})\operatorname{vec}(\mathbf{W}) $$ $$ = \frac{1}{\tau^2}\operatorname{vec}(\mathbf{W})^T\operatorname{vec}(\mathbf{W}_0)+\frac{1}{\sigma^2}\operatorname{vec}(\mathbf{T})^T\operatorname{vec}(\mathbf{WX}), $$
where $\otimes$ denotes the Kroenecker product.
I know that $$ \mathbf{V} \otimes \mathbf{U} = \left(\frac{1}{\tau^2}\mathbf{I}_q + \frac{1}{\sigma^2}\mathbf{X}\mathbf{X}^T\right)\otimes \mathbf{I}_D. $$
And $\mathbf{W}$ should somehow disappear out of the equation. $\mathbf{X}$, $\mathbf{T}$, $\mathbf{W}_0$, $\tau$ and $\sigma$ are known.
How can I solve for $\mathbf{M}$ (ideally unvectorized $\mathbf{M}$)?
ADDITIONAL INFORMATION: I know from my application that $\mathbf{V} \otimes \mathbf{U}$ is symmetric.