First, add git://git.gnome.org/libgsystem as a git submodule, e.g.:

$ git submodule add src/libgsystem git://git.gnome.org/libgsystem

Next, you'll need to have set up in your configure.ac:

PKG_CHECK_MODULES(GIO_UNIX, gio-unix-2.0 >= 2.34)

Then in your Makefile.am:

libgsystem_srcpath := src/libgsystem
libgsystem_cflags = $(GIO_UNIX_CFLAGS) -I$(srcdir)/src/libgsystem
libgsystem_libs = $(GIO_UNIX_LIBS)
include src/libgsystem/Makefile-libgsystem.am
noinst_LTLIBRARIES += libgsystem.la

It's very important to note here that libgsystem integrates with non-recursive automake. You can use it in recursive automake setups as well; it will just be partially non-recursive.

Finally, set up your application's CFLAGS and LIBADD to use it:

myapp_CFLAGS = $(AM_CFLAGS) -I$(srcdir)/src/libgsystem
myapp_LIBADD = libgsystem.la