  In 1972, Tutte posed the $3$-Flow Conjecture: that all $4$-edge-connected
graphs have a nowhere zero $3$-flow. This was extended by Jaeger et al.(1992)
to allow vertices to have a prescribed, possibly non-zero difference (modulo
$3$) between the inflow and outflow. They conjectured that all
$5$-edge-connected graphs with a valid prescription function have a nowhere
zero $3$-flow meeting that prescription (we call this the Strong $3$-Flow
Conjecture). Kochol (2001) showed that replacing $4$-edge-connected with
$5$-edge-connected would suffice to prove the $3$-Flow Conjecture and Lov\'asz
et al.(2013) showed that the $3$-Flow and Strong $3$-Flow Conjectures hold if
the edge connectivity condition is relaxed to $6$-edge-connected. Both problems
are still open for $5$-edge-connected graphs.
  The $3$-Flow Conjecture was known to hold for planar graphs, as it is the
dual of Gr\"otzsch's Colouring Theorem. Steinberg and Younger (1989) provided
the first direct proof using flows for planar graphs, as well as a proof for
projective planar graphs. Richter et al.(2016) provided the first direct proof
using flows of the Strong $3$-Flow Conjecture for planar graphs. We provide two
extensions to their result, that we developed in order to prove the Strong
$3$-Flow Conjecture for projective planar graphs.
