The parties in Cyprus should engage with the UN Secretary General`s envoy Jane Holl Lute in a creative and constructive manner, recognizing that the outcome will be critical in determining the future of the Cyprus negotiations, the UNSG`s Deputy Special Adviser and Special Representative in Cyprus Elizabeth Spehar said on Tuesday evening just a few days before Lute`s arrival to Cyprus, to meet with Cyprus President, Nicos Anastasiades, and Turkish Cypriot leader, Mustafa Akinci.