Let $X \in \mathbb{R}^d$ be a random variable with probability distribution $P$. Let $f:\mathbb{R}^d \to \mathbb{R}^d$ be an invertible function and let $P_{f}$ be the distribution of random variable $f(X)$. Suppose $P$ and $P_f$ are such that the mixture distribution $(1-\alpha)P + \alpha P_f$, for some $\alpha \in [0,1]$, is equal to the standard normal distribution $\mathcal{N}(0, I_{d\times d})$, where $I_{d\times d}$ is the $d\times d$ identity matrix.

Given $\alpha, f$, I'm interested in finding the distribution $P$ for which the above condition holds, *i.e.,* $(1-\alpha)P + \alpha P_f$ is equal to $\mathcal{N}(0, I_{d\times d})$. (By finding a distribution what I mean is that, at any given $x \in \mathbb{R}^d$, I would like to compute the probability density of $P$ at $x$.)

Here are my questions:

1) For any given $\alpha, f$, is there a unique $P$ which satisfies this *i.e.,* is the problem identifiable? Or can there be multiple distributions? Of course, when $\alpha = 1/2$ the problem is clearly not identifiable. Assuming $\alpha \neq 1/2$, is the problem identifiable?

2) Assuming the problem is identifiable, how can I compute the density of $P$ at any given $x \in \mathbb{R}^d$? We have the following relation between the densities of $P, P_f$ and $\mathcal{N}(0, I_{d\times d})$

$$\frac{1}{\sqrt{2\pi}} e^{-x^2/2} = (1-\alpha)p(x) + \alpha |\det{J_{f^{-1}}(x)}| p(f^{-1}(x)),$$

where $p(x)$ is the probability density of $P$ at $x$ and $\det{J_{f^{-1}}(x)}$ is the determinant of the Jacobian of $f^{-1}$ evaluated at $x$. Is there a way to compute $p(x)$ from this equation?