  The KPZ fixed point is a 2d random field, conjectured to be the universal
limiting fluctuation field for the height function of models in the KPZ
universality class. Similarly, the periodic KPZ fixed point is a conjectured
universal field for spatially periodic models. For both fields, their
multi-point distributions in the space-time domain have been computed recently.
We show that for the case of the narrow-wedge initial condition, these
multi-point distributions can be expressed in terms of so-called integrable
operators. We then consider a class of operators that include the ones arising
from the KPZ and the periodic KPZ fixed points, and find that they are related
to various matrix integrable differential equations such as coupled matrix mKdV
equations, coupled matrix NLS equations with complex time, and matrix KP-II
equations. When applied to the KPZ fixed points, our results extend previously
known differential equations for one-point distributions and equal-time,
multi-position distributions to multi-time, multi-position setup.
