dZ-Cluster tilting subcategories of singularity categories

For an exact category E with enough projectives and with a dZ-cluster tilting subcategory, we show that the singularity category of E admits a dZ-cluster tilting subcategory. To do this we introduce cluster tilting subcategories of left triangulated categories, and we show that there is a correspondence between cluster tilting subcategories of E and E. We also deduce that the Gorenstein projectives of E admit a dZ-cluster tilting subcategory under some assumptions. Finally, we compute the dZ-cluster tilting subcategory of the singularity category for a finite-dimensional algebra which is not Iwanaga-Gorenstein.