  We study the constrained minimum energy problem with an external field
relative to the $\alpha$-Riesz kernel $|x-y|^{\alpha-n}$ of order
$\alpha\in(0,n)$ for a generalized condenser $\mathbf A=(A_i)_{i\in I}$ in
$\mathbb R^n$, $n\geqslant 3$, whose oppositely charged plates intersect each
other over a set of zero capacity. Conditions sufficient for the existence of
minimizers are found, and their uniqueness and vague compactness are studied.
Conditions obtained are shown to be sharp. We also analyze continuity of the
minimizers in the vague and strong topologies when the condenser and the
constraint both vary, describe the weighted equilibrium vector potentials, and
single out their characteristic properties. Our arguments are based
particularly on the simultaneous use of the vague topology and a suitable
semimetric structure on a set of vector measures associated with $\mathbf A$,
and the establishment of completeness theorems for proper semimetric spaces.
The results remain valid for the logarithmic kernel on $\mathbb R^2$ and
$\mathbf A$ with compact $A_i$, $i\in I$. The study is illustrated by several
examples.
