Gauged supergravities which have a rich structure, typically including a gauged sigma model sector with an intricate potential, provide a furtile arena for searching novel exact solutions. These, in turn, may lead to discoveries of new types of string theory vacua. With these motivations in mind, we have studied the matter coupled chiral gauged supergravity in six dimensions. In this talk, we shall describe the necessary and sufficient conditions for the existence of Killing spinors in that theory and show that they imply most of the field equations. We also determine the remaining field equations and the general form of any supersymmetric solution. We then present an exact new supersymmetric dyonic string solution in which four hyperscalars that parametrize a quaternionic ball are mapped to internal space transverse to the string worldsheet. We show that the internal space is noncompact with finite volume, and it is a four dimensional analog of the Gell-Mann-Zweibach's two dimensional teardrop. We also find that the electric charge carried by the dyonic string is arbitrary, but its magnetic charge is fixed in Plankian units. Other salient properties of the solution, including its near horizon limit with enhanced supersymmetry, related work in the literature, open problems and future directions will also be discussed in this talk.