Suppose that there exists a random vector $\eta\equiv (\eta_1, \eta_2, \eta_3)$ continuously distributed on $\mathbb{R}^3$ and with full support. Can we always find a vector $\epsilon\equiv (\epsilon_1,\epsilon_2,\epsilon_3)$ continuously distributed on $\mathbb{R}^3$ and with full support such that $$ \eta_1\equiv \epsilon_1-\epsilon_3\\ \eta_2\equiv \epsilon_1-\epsilon_2\\ \eta_3\equiv \epsilon_2-\epsilon_3\\ $$

If yes, how can we show it? If not, can you provide a counterexample?