Yes — the Social System also performs Integration, but in a different sense from the “Integration (I)” that belongs to Parsons’ four major subsystems.

This confusion happens because Integration appears at two levels in Parsons’ theory:

✅ Level 1: Integration (I) as one of the FOUR AGIL functions of the WHOLE SOCIETY

At the societal level:

AGIL Function	Subsystem Responsible
A	Economy
G	Polity
I	Social System (but mainly legal-normative order)
L	Social + Cultural System

Here, Integration (I) is a macro-function that manages:

social order
regulation of conflict
legal rules
normative coordination across subsystems

The main integrators are:

law
judiciary
shared norms
institutionalised rules
constitution
bureaucracy as normative organisation

So at this macro level, “Integration = the function of maintaining social order”.