We investigate properties of the Euler system associated to certain automorphic representations of the unitary similitude group GU(2,1) with respect to an imaginary quadratic field E, constructed by Loeffler-Skinner-Zerbes. By adapting Mazur and Rubin's Euler system machinery we prove one divisibility of the "rank 1" Iwasawa main conjecture under some mild hypotheses. When p is split in E we also prove a "rank 0" statement of the main conjecture, bounding a particular Selmer group in terms of a p-adic distribution conjecturally interpolating complex L-values. We then prove descended versions of these results, at integral level, where we bound certain Bloch-Kato Selmer groups. We will also discuss the case where p is inert, which is a work in progress.