No nonnegative solution of $\int \frac{x(t)\,dt}{(y-t)^2+1}=e^{-y^2}$

Prove that the equation

\[ \int _ { - \infty } ^ { \infty } { \frac { x ( t ) d t } { ( y - t ) ^ { 2 } + 1 } } = e ^ { - y ^ { 2 } }\tag{44} \]

has no nonnegative solutions \(x ( t )\)