  We propose a simple method of synchronous combined modulations to generate
the exact analytic solutions for a spin-orbit (SO) coupled ultracold atom held
in a non-Hermitian double-well potential. Based on the obtained analytical
solutions, we mainly study the parity-time ($\mathcal{PT}$) symmetry of this
system and the system stability for both balanced and unbalanced gain-loss
between two wells. Under balanced gain and loss, the effect of the proportional
constants between synchronous combined modulations and the SO-coupling strength
on the $\mathcal{PT}$-symmetry breaking is revealed analytically. Surprisingly,
we find when the Zeeman field is present, the stable spin-flipping tunneling
between two wells can not occur in the non-Hermitian SO-coupled ultracold
atomic system, but the stable spin-conserving tunneling can be performed. Under
unbalanced gain and loss, the unique set of parameter conditions that can cause
the system to stabilize is found. The results may provide a possibility for the
exact control of $\mathcal{PT}$-symmetry breaking and quantum spin dynamics in
a non-Hermitian SO-coupled system.
