I'm studying a dynamical system with $\mathbf{D}_{3}$ symmetry (the symmetry group of an equilateral triangle), which is given by:
$\begin{align*} d\mathbf{x}_{0}/dt &= \mathbf{f}(\mathbf{x}_{2}, \mathbf{x}_{0}, \mathbf{x}_{1}) \\ d\mathbf{x}_{1}/dt &= \mathbf{f}(\mathbf{x}_{0}, \mathbf{x}_{1}, \mathbf{x}_{2}) \\ d\mathbf{x}_{2}/dt &= \mathbf{f}(\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{0}) \end{align*}$,
where $\mathbf{x}_{0}, \mathbf{x}_{1}, \mathbf{x}_{2} \in \mathbb{R}^{k}$ and $\mathbf{f}(\mathbf{u}, \mathbf{v}, \mathbf{w}) = \mathbf{f}(\mathbf{w}, \mathbf{v}, \mathbf{u})$.
Is a fixed point at $(\mathbf{x}_{0}, \mathbf{x}_{1}, \mathbf{x}_{2}) = (\mathbf{0}, \mathbf{0}, \mathbf{0})$ guaranteed by the symmetry in the problem?