Wild automorphisms of graded algebras

In 2004, Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a $\mathbb{Z}$-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that cannot be decomposed onto elementary automorphisms respecting the grading. We describe all gradings allowing graded-wild automorphisms. We also discuss systems of automorphisms generating the group of graded automorphisms