  We investigate the existence of solutions with 16 residual supersymmetries to
Type IIB supergravity on a space-time of the form $AdS_6 \times S^2$ warped
over a two-dimensional Riemann surface $\Sigma$. The $SO(2,5) \times SO(3)$
isometry extends to invariance under the exceptional Lie superalgebra $F(4)$.
In the present paper, we construct the general Ansatz compatible with these
symmetries, derive the corresponding reduced BPS equations, and obtain their
complete local solution in terms of two locally holomorphic functions ${\cal
A}_\pm$ on $\Sigma$, subject to certain positivity and regularity conditions.
Globally, $({\cal A}_+, {\cal A}_-)$ are allowed to be multiple-valued on
$\Sigma$ and be holomorphic sections of a holomorphic bundle over $\Sigma$ with
structure group contained in $SU(1,1) \times { C}$. Globally regular solutions
are expected to provide the near-horizon geometry of $(p,q)$ 5-brane and
7-brane webs which are holographic duals to five-dimensional conformal field
theories. A preliminary analysis of the positivity and regularity conditions
will be presented here, leaving the construction of globally regular solutions
to a subsequent paper.
