Let $\nu(x), \mu(x), \sigma(x)$ be real smooth functions, where $\nu(x)$ is a density function (i.e. $\int_{-\infty}^\infty\nu(x)dx=1$,$\nu(x)\geq 0$ ) and $\sigma(x)>0\forall x\in\mathbb{R}$. Does there exist $\tilde{\nu}, \tilde{\mu}, \tilde{\sigma}$ smooth functions, where $\tilde{\nu}$ is a density function and $\tilde{\sigma}(x)>0 \forall x\in\mathbb{R}$, such that $$ \nu(\frac{y-\mu(x)}{\sigma(x)}) = \tilde{\nu}(\frac{x-\tilde{\mu}(y)}{\tilde{\sigma}(y)}) $$ for all $x,y\in\mathbb{R}$?
Potentially, we can assume additional assumptions for $\nu, \mu, \sigma,\tilde{\nu}, \tilde{\mu}, \tilde{\sigma}$ such as Lipchitzity or any similiar "not much restrictive" assumptions if that helps.