On probability density functions which are their own characteristic functions

Let $p$ be the probability density of a probability distribution $P$ on the real line $\mathbf R$ with respect to the Lebesgue measure. The characteristic function $\widehat p$ of $p$ is defined as
$$ \widehat p(x):=\int_{\mathbf{R}}e^{ixy}p(y)\,dy,\qquad x\in\mathbf{R}. $$
We consider probability densities $p$ which are their own characteristic functions, that means
\begin{equation} \widehat p(x)=\frac1{p(0)}p(x),\qquad x\in\mathbf{R}. \tag{1} \end{equation}
By linear combination of Hermitian functions we find a family of probability densities which are solutions of this integral equation. These solutions are entire functions of order 2 and type $\frac12$. This is contradictory to Corollary 3 in [J. L. Teugels, Bull. Soc. Math. Belg., 23 (1971), pp. 236–262.]. Furthermore, we characterize the general solution of the integral equation (1) within the convex cone of probability density functions.