  We present two types of systems of differential equations that can be derived
from a set of discrete integrable systems which we call the closed geometric
crystal chains. One is a kind of extended Lotka-Volterra systems, and the other
seems to be generally new but reduces to a previously known system in a special
case. Both equations have Lax representations associated with what are known as
the loop elementary symmetric functions, which were originally introduced to
describe products of affine type A geometric crystals for symmetric tensor
representations. Examples of the derivations of the continuous time Lax
equations from a discrete time one are described in detail, where a novel
method of taking a continuum limit by assuming asymptotic behaviors of the
eigenvalues of the Lax matrix in Puiseux series expansions is used.
