  We derive the necessary and sufficient conditions for the existence of a
Killing spinor in N=(1,0) gauge supergravity in six dimensions coupled to a
single tensor multiplet, vector multiplets and hypermultiplets. These are shown
to imply most of the field equations and the remaining ones are determined. In
this framework, we find a novel 1/8 supersymmetric dyonic string solution with
nonvanishing hypermultiplet scalars. The activated scalars parametrize a 4
dimensional submanifold of a quaternionic hyperbolic ball. We employ an
identity map between this submanifold and the internal space transverse to the
string worldsheet. The internal space forms a 4 dimensional analog of the
Gell-Mann-Zwiebach tear-drop which is noncompact with finite volume. While the
electric charge carried by the dyonic string is arbitrary, the magnetic charge
is fixed in Planckian units, and hence necessarily non-vanishing. The source
term needed to balance a delta function type singularity at the origin is
determined. The solution is also shown to have 1/4 supersymmetric AdS_3 x S^3
near horizon limit where the radii are proportional to the electric charge.
