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?