An Exactly Soluble Three-Dimensional Toy Model with a Saddle Separatrix

We present a three-dimensional autonomous ODE system that admits an exact closed-form solution in its transition coordinate and explicit exponential bounds in the transverse coordinates. The system possesses a hyperbolic saddle whose two-dimensional stable manifold is an invariant plane that separates the phase space into two basins of attraction. The model is not derived from the Navier-Stokes equations. It is a toy system whose only purpose is to provide a trivially verifiable benchmark for edge-tracking algorithms in hydrodynamic stability. The saddle's eigenvalues are chosen to match the logarithmic contraction and expansion rates of the Jonah triangle. This parameter choice is noted but carries no dynamical derivation.