Let $M$ be a toric Kahler manifold with $\text{dim}_{\mathbb{R}} = 4$. Let $V$ be a Killing vector field associated to the action of the two-torus $\mathbb{T}^2$ on $M$. We also assume the existence of a metric $\langle, \rangle$ and denote the Killing one-form $v = \langle V, \cdot \rangle$. I'm studying the differential equation $$ \phi (dv)^+ = -2 (1+V^2) (d\phi \wedge v)^+, \quad \phi : M \to \mathbb{R}. $$ ${}^+$ denotes self-duality where conventions are such that the symplectic form $\omega$ is anti self-dual, i.e. $\omega = \omega^-$. (In detail: $\alpha^\pm = \frac{1}{2} (\alpha \pm \star \alpha)$, $\star\alpha_{ab} = -\frac{1}{2} \epsilon_{abcd} \alpha^{cd}$.)
Clearly $\phi = 0$ is a solution. I want to show that this is the only solution.
Remarks: (1) The only thing charged under $V$ in the above equation is $\phi$. If one can assume the Lie derivative to vanish: $\mathcal{L}_V \phi = \nabla_V \phi = 0$, contraction of the above with $V$ leads to $$ V^2 d\phi = \frac{1}{(1+V^2)} \phi i_V (dv)^+ $$ where $i_v$ denotes interior multiplication.
(2) I studied the equation explicitly for $\mathbb{CP}^2$ and $\mathbb{CP}^1 \times \mathbb{CP}^1$. Apart from the trivial solution one finds a singular one whose gradient $d\phi$ connects the "poles" $V^2 = 0$. On $\mathbb{CP}^2$, $dv^+ = 0$ vanishes at one of the poles and $d\phi = 0$ here. The solution diverges at the other pole however. On $\mathbb{CP}^1 \times \mathbb{CP}^1$ the non-trivial solution diverges at both poles (which in general are $\mathbb{CP}^1$s). This time, one has $dv^+ = 0$ on the equator. Note that in both examples the non-trivial solutions diverge at "poles" that are $\mathbb{CP}^1$s; moreover, in both examples the sole critical points of $\phi$ are those where $dv^+ = 0$. Finally, in both examples $\nabla_V \phi = 0$ follows from the differential equations.
(3) One can of course consider the moment map $\mu : M \to \mathfrak{t}^*$, which induces a map $\mu^V : M \to \mathbb{R}, p \mapsto \langle \mu(p), V \rangle$. $\mathfrak{t}$ is the Lie algebra of the torus, $\langle, \rangle$ denotes the usual pairing with $\mathfrak{t}^*$. One can rephrase the above via $d\mu^V = i_V \omega$.
(4) If $p \in M$ is a critical point of $\phi$, $d\phi(p) = 0$ it follows that $dv^+(p) = 0$. In the $\mathbb{CP}^2$ example, only one such point exists. I have tried to argue that there is generally only one point on $M$ where $dv^+$ vanishes, thus $\phi$ has at most one critical point, which (with additional assumptions) would imply that no such smooth solutions exists.