Support, absolute continuity and harmonic moments of fixed points of the multivariate smoothing transform

Consider the multivariate smoothing transform fixed-point equation: $\eta =$ law of $\sum_{i=1}^N A_i Z_i$, where $N \geq 0$ is a random integer, $(A_i)_{i \geq 1}$ are $d \times d$ random nonnegative matrices, $(Z_i)_{i \geq 1}$ is a sequence of $\mathbb{R}_+^d$-valued random variables independent of $(N, A_1, A_2, \cdots)$, and all $Z_i$ have the same law $\eta$. For each fixed point $\eta$, under suitable conditions, we describe its support, establish its absolute continuity, and prove the existence of its harmonic moments.