  We consider representation varieties in $SL_2$ for lattices in solvable Lie
groups, and representation varieties in $sl_2$ for finite-dimensional Lie
algebras. Inside them, we examine depth 1 characteristic varieties for
solvmanifolds, respectively resonance varieties for cochain Differential Graded
Algebras of Lie algebras. We prove a general result that leads, in both cases,
to the complete description of the analytic germs at the origin, for the
corresponding embedded rank 2 jump loci.
