1. The real roots of the equation $\frac{1}{x^{2}+1}+\frac{x^{2}+1}{x^{2}}=\frac{10}{3 x}$ are
$=1 . \frac{3 \pm \sqrt{5}}{2}$.
The original equation is transformed into
$\frac{x}{x^{2}+1}+\frac{x^{2}+1}{x}=\frac{10}{3}$.
Let $\frac{x}{x^{2}+1}=y$, equation (1) becomes
$$
y+\frac{1}{y}=3+\frac{1}{3} \text {. }
$$

Then $y_{1}=3$ or $y_{2}=\frac{1}{3}$.
Substituting, we get $\frac{x}{x^{2}+1}=3$ or $\frac{x}{x^{2}+1}=\frac{1}{3}$.
Simplifying, we get
$3 x^{2}-x+3=0$ (no real solutions) or $x^{2}-3 x+1=0$.
Solving, we get $x=\frac{3 \pm \sqrt{5}}{2}$.
Upon verification, these are solutions to the original equation.