Given an edge-capacitated graph and kterminal vertices, the maximum integer multiterminal flow problem (MaxIMTF) is to route the maximum number of flow units between the terminals. For directed graphs, we introduce a new parameter kL ≤ k and prove that MaxIMTF is $\mathcal{NP}$-hard when k = kL = 2 and when kL = 1 and k = 3, and polynomial-time solvable when kL = 0 and when kL = 1 and k = 2. We also give an 2 log2 (kL + 2)-approximation algorithm for the general case. For undirected graphs, we give a family of valid inequalities for MaxIMTF that has several interesting consequences, and show a correspondence with valid inequalities known for MaxIMTF and for the associated minimum multiterminal cut problem.