  We consider a family of holomorphic PDEs whose singular locus is given by the
zero set of an analytic map $P$ with $P(0)=0$. Our goal is to establish
conditions for the existence and uniqueness of formal power series solutions
and to determine their divergence rate. In fact, we prove that the solution is
Gevrey in $P$, giving new information on divergency while compared to the
classical Gevrey classes. If $P$ is not singular at $0$, we also provide
Poincar\'e conditions to recover convergent solutions. Our strategy is to
extend the dimension and lift the given PDE to a problem where results of
singular PDEs can be applied. Finally, examples where the Gevrey class in $P$
is optimal are included.
