  We study J\'anossy densities of a randomly thinned Airy kernel determinantal
point process. We prove that they can be expressed in terms of solutions to the
Stark and cylindrical Korteweg-de Vries equations; these solutions are Darboux
tranformations of the simpler ones related to the gap probability of the same
thinned Airy point process. Moreover, we prove that the associated wave
functions satisfy a variation of Amir-Corwin-Quastel's integro-differential
Painlev\'e II equation. Finally, we derive tail asymptotics for the relevant
solutions to the cylindrical Korteweg-de Vries equation and show that they
decompose asymptotically into a superposition of simpler solutions.
