I was going through a proof in some integrable systems lecture notes about the relationship between lax pairs and zero curvature. The proof starts as follows:
Let $ Lf = \lambda f $, where $ L $ is a self-adjoint linear operator, $ \lambda (t)<0 $ is a simple eigenvalue, and $ f(x,t) $ is a function. Suppose also that $ L_t=LA-AL=[L, A] $, where the subscript is a partial derivative with respect to time $ t $, and $ A $ is another linear operator. From the proof of the isospectral flow theorem, we have $ 0=\lambda_t f= (L-\lambda )(f_t+Af) $, and hence $ f_t+Af=c(t)f $ for some function $ c(t) $ which is independent of $ x $, We can then apply an integrating factor to deduce that there exists $ \hat{f}(x,t) $ which satisfies $ L\hat{f}=\lambda \hat{f} $ and $\hat{f}_t + A\hat{f}=0 $...
I can't see how using an integrating factor gives us this $ \hat{f} $. I've been able to obtain $$ \exp\left(\int^t \left(A(t')-c(t')\right) \mathrm{d}t'\right)f(x,t)=h(x) $$ for some function $ h(x) $, but I'm not sure how to proceed from here.