It is shown that for every annulus
P
=
{
z
∈
C
n
:
δ
>
|
z
|
>
r
}
P = \{ z \in {{\mathbf {C}}^n}:\delta > |z| > r\}
,
δ
>
0
\delta > 0
, there exists a set-valued correspondence
z
→
K
(
z
)
:
P
→
2
C
n
z \to K(z):P \to {2^{{{\mathbf {C}}^n}}}
, whose graph is a bounded relatively closed subset of the manifold
{
(
z
,
w
)
∈
P
×
C
n
:
z
1
w
1
+
⋯
+
z
n
w
n
=
1
}
\{ (z,w) \in P \times {{\mathbf {C}}^n}:{z_1}{w_1} + \cdots + {z_n}{w_n} = 1\}
which can be covered by
n
n
-dimensional analytic manifolds. The analytic set-valued selection
K
K
obtained thereby is then applied to several problems in complex analysis and spectral theory which involve solving the equation
a
1
x
1
+
⋯
+
a
n
x
n
=
y
{a_1}{x_1} + \cdots + {a_n}{x_n} = y
. For example, an elementary proof is given of the following special case of a theorem due to Oka: every bounded pseudoconvex domain in
C
2
{{\mathbf {C}}^2}
is a domain of holomorphy.