  We study a second-order linear differential equation known as the deformed
cubic oscillator, whose isomonodromic deformations are controlled by the first
Painlev{\'e} equation. We use the generalised monodromy map for this equation
to give solutions to the infinite-dimensional Riemann-Hilbert problems arising
from the Donaldson-Thomas theory of the A2 quiver. These are the first known
solutions to such problems beyond the uncoupled case. The appendix by Davide
Masoero contains a WKB analysis of the asymptotics of the monodromy map.
