Consider the following system for $x_1$, $x_2$, $x_3$ positive: $$\frac{dx_{i}}{dt}=-x_{i}\left(x_{i}-\bar{x}\right)\qquad\text{where}\ \bar{x}=\frac{x_1+x_2+x_3}{3}$$ Given a starting point such that $x_1x_2x_3=1$, it is easy to show that the system remains on the surface $x_1x_2x_3=1$ and heads towards the stationary point $x_1=x_2=x_3=1$. Is it possible to find an analytic solution, so that one could calculate where the system is at time $t$?
I've noted that the system is in generalized Lotka-Volterra form, and I've been wondering whether the system travels on a geodesic, but I don't know how to show this, or whether it would be useful.
Running a few numerical solutions, it appears that if x3>x2>x1, this always remains the case, and that x3 monotonically decreases, x1 monotonically increases, but the time derivative of x2 may change sign.