$$\begin{align}\dot{x}_{0}&=- \frac{x_{0} \cdot x_{2}}{\sqrt{x_{0}^{2} + x_{1}^{2}}} - x_{1}\\\dot{x}_{1}&=x_{0} - \frac{x_{1} \cdot x_{2}}{\sqrt{x_{0}^{2} + x_{1}^{2}}}\\\dot{x}_{2}&=\frac{x_{0}}{\sqrt{x_{0}^{2} + x_{1}^{2}}}\end{align}$$
The integral surface of a torus. Link to DETEST description
$$\begin{align}x_{0}(0)&=3.0\\x_{1}(0)&=0.0\\x_{2}(0)&=0.0\end{align}$$
No exact solution provided.
[untested]
$$\begin{align}x_{0}(0)&\subset~\left[ 3.0, 3.25\right]\\x_{1}(0)&=0.0\\x_{2}(0)&=0.0\end{align}$$
* rounded (non-rigorously) to 5 decimal places to preserve screen real estate.