I want to find a "true" nonlinear least squares problem which does have an analytical solution.
I tried to construct something with a Dirac-Delta function and ended up with $y_n = c^2\delta(x_n-x_1)x_n+\varepsilon_n$, in which I assumed a dataset $\mathcal{D}=\left\{(x_1,y_1),\ldots,(x_N,y_N) \right\}$. This equation is not really nonlinear in the coefficients as we can reformulate the regression equation to $y_n=\tilde{c}\delta(x_n-x_1)x_n+\varepsilon_n$ with $\tilde{c}\geq 0$, hence this would still count as a linear regression with nonlinear basis functions.
Is there a well-known example of a nonlinear least squares problem that does have an analytical closed-form solution? References are appreciated.