  We perform a systematic analysis of flow-like solutions in theories of
Einstein gravity coupled to multiple scalar fields, which arise as holographic
RG flows as well as in the context of cosmological solutions driven by scalars.
We use the first order formalism and the superpotential formulation to classify
solutions close to generic extrema of the scalar potential, and close to
"bounces," where the flow is inverted in some or all directions and the
superpotential becomes multi-valued. Although the superpotential formulation
contains a large redundancy, we show how this can be completely lift by
suitable regularity conditions. We place the first order formalism in the
context of Hamilton-Jacobi theory, where we discuss the possibility of
non-gradient flows and their connection to non-separable solutions of the
Hamilton-Jacobi equation. We argue that non-gradient flows may be useful in the
presence of global symmetries in the scalar sector.
