Gradient flow for a Morse function on the flat torus

Differentiating gives $df=-\sin \theta d\theta -\sin \varphi d\varphi$, hence the displayed negative-gradient equations. Their zeros give exactly the four listed critical points, and the diagonal Hessian $\mathrm{diag}(-\cos \theta ,-\cos \varphi )$ gives the stated indices.

The circles $\theta \in \{0,\pi \}$ and $\varphi \in \{0,\pi \}$ are invariant and supply the coordinate-circle separatrices. On either component of $S^1\setminus \{0,\pi \}$, each nonconstant coordinate trajectory has backward limit $0$ and forward limit $\pi$ (it increases on $(0,\pi )$ and decreases on $(\pi ,2\pi )$ in the displayed coordinate). Thus the trajectories off those circles run from the maximum to the minimum in four arc-choice components. In each component, quotienting the two integration constants by common time translation leaves a one-parameter family.