  A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be
contracted to $H$. We prove that for a positive integer $\ell$ and a non-empty
planar graph $H$ with at least $\ell-1$ connected components, there exists a
function $f_{H, \ell}:\mathbb{N}\rightarrow \mathbb{R}$ satisfying the property
that every graph $G$ with a family of vertex subsets $Z_1, \ldots, Z_m$
contains either $k$ pairwise vertex-disjoint minor-models of $H$ each
intersecting at least $\ell$ sets among prescribed vertex sets, or a vertex
subset of size at most $f_{H, \ell}(k)$ that meets all such minor-models of
$H$. This function $f_{H, \ell}$ is independent with the number $m$ of given
sets, and thus, our result generalizes Mader's $\mathcal S$-path Theorem, by
applying $\ell=2$ and $H$ to be the one-vertex graph. We prove that such a
function $f_{H, \ell}$ does not exist if $H$ consists of at most $\ell-2$
connected components.
