I suspect the following probability distribution may have a name.
Let $X_j \sim N^d ( \mu_j , \Sigma_j )$ for $j = 1,2$. ($X_1$ and $X_2$ are not necessarily independent.) Let $a \sim \mathrm{Uni} (0,1)$ be independent of $X_1$ and $X_2$. The marginal of the posterior distribution $p(X_1,X_2,a|X_1=aX_2)$ is possible to calculate in closed-form. I.e. the distribution
$$ p(a) = C \int_{\mathbb{R}^d} f (X , a X) d X , \tag{1}$$
where $f(X_1,X_2)$ is the joint pdf of $X_1$ and $X_2$, and $C$ is a normalization constant, has a relatively simple closed-form. In particular, for the case $d = 1$ and with $X_1$ and $X_2$ independent, this reads
$$ p(a) = C \sqrt{\frac{2\pi \sigma_1^2 \sigma_2^2}{\sigma_2^2 + \sigma_1^2 a^2 }} \exp{ \left\lbrace \frac{\left( \sigma_2^2 \mu_1 + \sigma_1^2 \mu_2 a \right)^2}{2 \sigma_1^2 \sigma_2^2 (\sigma_2^2 + \sigma_1^2 a^2)} \right\rbrace} . \tag{2} $$
(A general expression is possible for $d > 1$, but I am too lazy to write it down.)
In other words, the random vector $(X_1 , X_2) \in \mathbb{R}^{2d}$ is known to lie in the subspace given by $\{ X_1 = a X_2 \}$ and (1) is the density that expresses how likely this as a function of $a$ assuming no prior knowledge of $a$ (i.e., that it is uniformly distributed).
Question: Does the distribution in (1)-(2) have a name? This question of conditioning a Gaussian density to lie in a subspace must be considered somewhere in the literature.