  We derive differential equations for multiplicative statistics of the Bessel
determinantal point process depending on two parameters. In particular, we
prove that such statistics are solutions to an integrable nonlinear partial
differential equation describing isospectral deformations of a Sturm--Liouville
equation. We also derive identities relating solutions to the integrable
partial differential equation and to the Sturm--Liouville equation which imply
an analogue for Painlev\'e V of Amir--Corwin--Quastel ``integro-differential
Painlev\'e II equation''. This equation reduces, in a degenerate limit, to the
system of coupled Painlev\'e V equations derived by Charlier and Doeraene for
the generating function of the Bessel process, and to the Painlev\'e V equation
derived by Tracy and Widom for the gap probability of the Bessel process.
Finally, we study an initial value problem for the integrable partial
differential equation. The approach is based on Its--Izergin--Korepin--Slavnov
theory of integrable operators and their associated Riemann--Hilbert problems.
