  Inspired by G. Frieden's recent work on the geometric R-matrix for affine
type A crystal associated with rectangular shaped Young tableaux, we propose a
method to construct a novel family of discrete integrable systems which can be
regarded as a geometric lifting of the generalized periodic box-ball systems.
By converting the conventional usage of the matrices for defining the Lax
representation of the discrete periodic Toda chain, together with a clever use
of the Perron-Frobenious theorem, we give a definition of our systems. It is
carried out on the space of real positive dependent variables, without
regarding them to be written by subtraction-free rational functions of
independent variables but nevertheless with the conserved quantities which can
be tropicalized. We prove that, in this setup an equation of an analogue of the
`carrier' of the box-ball system for assuring its periodic boundary condition
always has a unique solution. As a result, any states in our systems admit a
commuting family of time evolutions associated with any rectangular shaped
tableaux, in contrast to the case of corresponding generalized periodic
box-ball systems where some states did not admit some of such time evolutions.
