The qualitative behavior of a recently formulated ODE model for the dynamics of heterogenous aggregates is analyzed. Aggregates contain two types of particles, oligomers and cross-linkers. The motivation is a preparatory step of cellular autophagy, the aggregation of oligomers of the protein p62 in the presence of ubiquitin cross-linkers. A combination of explicit computations, formal asymptotics, and numerical simulations has led to conjectures on the bifurcation behavior, certain aspects of which are proven rigorously in this work. In particular, the stability of the zero state, where the model has a smoothness deficit is analyzed by a combination of regularizing transformations and blow-up techniques. On the other hand, in a different parameter regime, the existence of polynomially growing solutions is shown by Poincar\'e compactification, combined with a singular perturbation analysis .