On the division problem for a tempered distribution that depends holomorphically on a parameter

We give sufficient conditions ensuring a construction of solution to the equation
$$ \sum^m_{k=0}P_k(\sigma)\lambda^ku(\lambda)=f(\lambda), $$
with $\sigma\in\mathbb R^n$ and $\lambda\in G\subset\mathbb C$, where $f(\lambda) and u(\lambda)$are tempered distributions depending holomorphically on $\lambda$, while the polynomial $P_m(\sigma)$ may have real zeros.