1. The sum of all real numbers $x$ that satisfy the equation $\sqrt{3 x-4}+\sqrt[3]{5-3 x}=1$ is $\qquad$ .
II. $1 . \frac{22}{3}$.
Let $\sqrt{3 x-4}=t$, then $t \geqslant 0$.
The original equation becomes $1-t=\sqrt[3]{1-t^{2}}$, that is,
$t(t-1)(t-3)=0$.
Therefore, $t=0,1,3$.
From $t=0$, we get $x=\frac{4}{3}$; from $t=1$, we get $x=\frac{5}{3}$; from $t=3$, we get $x=\frac{13}{3}$. After verification, all are solutions to the equation, and the sum is $\frac{22}{3}$.