The Dubai meetings are the first, fresh Afghan-initiated efforts to restart peace talks aimed at bringing a negotiated end to the conflict ahead of the final withdrawal of international combat troops due at the end of this year.
The Taliban has denied links to the faction, organised by former Taliban finance minister Aga Jan Mohtism.
The High Peace Council said in a statement today that the delegation it met with clearly had indicated they were ready for peace talks and that both sides agreed on the need for further dialogue, both inside and outside of Afghanistan.