  We consider the six-dimensional dynamical system in three components
introduced by Ryan to describe the scenario of Belinskii, Khalatnikov and
Lifshitz to the cosmological singularity when the spatial metric tensor is not
diagonal. Despite its nonintegrability, recently proven by Goldstein and
Piechocki, the three four-dimensional systems defined by canceling one of the
three components happen to be integrable. We express their general solution as
a rational function of, respectively, two exponential functions, a third
Painlev\'e function, two exponential functions.
