1. Given that $a$ is an integer, the equation concerning $x$
$$
\frac{x^{2}}{x^{2}+1}-\frac{4|x|}{\sqrt{x^{2}+1}}=a-3
$$

has real roots. Then the possible values of $a$ are
二、1.1,2,3.
From the original equation, we get
$$
\left(\frac{|x|}{\sqrt{x^{2}+1}}-2\right)^{2}=1+a \text {. }
$$

Also, from $0 \leqslant \frac{|x|}{\sqrt{x^{2}+1}}<1$, we know
$$
1<\left(\frac{|x|}{\sqrt{x^{2}+1}}-2\right)^{2} \leqslant 4,
$$

which implies $1<1+a \leqslant 4 \Rightarrow 0<a \leqslant 3$.
Therefore, the possible values of $a$ are $1,2,3$.