  The equations underlying all supersymmetric solutions of six-dimensional
minimal ungauged supergravity coupled to an anti-self-dual tensor multiplet
have been known for quite a while, and their complicated non-linear form has
hindered all attempts to systematically understand and construct BPS solutions.
In this paper we show that, by suitably re-parameterizing these equations, one
can find a structure that allows one to construct supersymmetric solutions by
solving a sequence of linear equations. We then illustrate this method by
constructing a new class of geometries describing several parallel spirals
carrying D1, D5 and P charge and parameterized by four arbitrary functions of
one variable. A similar linear structure is known to exist in five dimensions,
where it underlies the black hole, black ring and corresponding microstate
geometries. The unexpected generalization of this to six dimensions will have
important applications to the construction of new, more general such
geometries.
