I am trying to solve the equation $ $ \mathrm{d}G(x,y) = \mathrm{Vol}(x)+(-1)^{n+1}\mathrm{Vol}(y):= H $ $ for $ G\in \Omega^{n-1}\bigl((\mathbb{S}^n\times \mathbb{S}^n)\backslash \Delta=:M\bigr)$ . Here $ \mathrm{Vol}$ is the standard volume form on $ \mathbb{S}^n$ and $ \Delta$ the diagonal. The solution for $ n=1$ is the oriented angle $ \alpha(x,y)$ from $ x$ to $ y$ up to a sign (depends on the orientation convention), i.e. $ $ G(x,y)= \pm \alpha(x,y)\quad \text{for all }x,y\in \mathbb{S}^1. $ $ I can find a solution for general $ n\in \mathbb{N}$ as $ $ G = \int_{[0,1]} \psi^*H $ $ for a contraction $ \psi: [0,1]\times M \rightarrow M$ of $ M$ onto the antidiagonal $ \overline{\Delta}:=\{(x,-x)\}$ (on $ \overline{\Delta}$ holds namely $ H=0$ ). However, the solutions I obtain (in coordinates $ x^i$ on $ \mathbb{R}^{n+1}$ ) look too complicated and I can not work with them further.

Does anybody have any idea how to solve this equation “nicely” e.g. by some natural construction of $ G(x,y)$ similar to the case $ n=1$ ?

I am mainly interested in the case $ n=3$ . Can the Lie group structure be used to construct a solution?

Thanks for any ideas!

(Bigger picture: $ H$ is a smooth integral kernel of the orthogonal projection to harmonic forms and $ G$ the “Green kernel” of $ \mathrm{d}$ )