Non-vanishing of vector-valued Poincar\'e series

We prove a vector-valued version of Mui\'c's integral non-vanishing criterion for Poincar\'e series on the upper half-plane $\mathcal H$. Moreover, we give an accompanying result on the construction of vector-valued modular forms in the form of Poincar\'e series. As an application of these results, we construct and study the non-vanishing of the classical and elliptic vector-valued Poincar\'e series.