Let $(\Bbb R^5,(x,y,z,w,t))$ be a smooth manifold and $\alpha_i$, $i\in\{1,\cdots,5\}$ five differential $1$-form such that $d \alpha_5 =\alpha_1\wedge\alpha_4+\alpha_2\wedge\alpha_3$. Suppose $e_i$ be the dual vector of the $1$-form $\alpha_i$, for any $i \in \{1,\cdots, 5\}$ and $$g=\alpha_1\otimes \alpha_1+\alpha_2\otimes \alpha_2+\alpha_3\otimes \alpha_3+\alpha_4\otimes \alpha_4+\alpha_5\otimes \alpha_5,$$ be Riemannian metric and denote by $R$ its curvature tensor. Then
how to solve following PDE? $$R(e_i,e_5)e_5=f(x,y,z,w,t)e_i,\qquad i\neq5,$$
for some smooth non-constant function $f$ independent of $e_i$.
A trivial solution ($f=0$) is as follow: $$\alpha_1=dx,\ \alpha_2=dy,\ \alpha_3=dw+tdy,\ \alpha_4=dz+tdx,\ \alpha_5=dt-zdx-wdy.$$
Can be deform this solution to smooth non-constant function $f$?