Consider the equality: $$ \mathbf{g}(\mathbf{x}, t)\mathbf{g}(\mathbf{x}, t)^\top \nabla_\mathbf{x} H(\mathbf{x}, t)^\top = - \frac{1}{2} \nabla_\mathbf{x} \cdot \left( \mathbf{G}(\mathbf{x}, t)\mathbf{G}(\mathbf{x}, t)^\top \right), $$ where $\mathbf{g}: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^{n \times n}$, $\mathbf{G}: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^{n \times n}$, $\mathbf{H}: \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^{n}$, and $\nabla_\mathbf{x} \cdot$ is the operator for the divergence of a matrix.
How can I solve for $\mathbf{g}(\mathbf{x}, t)\mathbf{g}(\mathbf{x}, t)^\top)$.