 Solution

Normalize the second price to one and write $p>0$ for the first price and $t>0$ for the varying endowment. Let $Z(p,t)$ be first-good aggregate excess demand. Under Walras law, the second excess-demand coordinate equals $-pZ(p,t)$, so one market-clearing equation suffices. A regular exchange economy at fixed $t$ has $Z_p(p,t)\ne 0$ at every equilibrium.

A sufficient joint condition is that zero be a regular value of the smooth map $Z:(0,\infty {)}^2\to \mathbb{R}$:To prove the almost-everywhere conclusion, the regular level set theorem makes $M=Z^{-1}(0)$ a smooth one-dimensional manifold. Project $M$ to the parameter axis by $\pi (p,t)=t$. Its tangent space consists of vectors $(h,k)$ with $Z_{p}h+Z_{t}k=0$. If $Z_p\ne 0$, every $k$ is possible, so the derivative of $\pi$ is surjective. If $Z_p=0$, the joint condition gives $Z_t\ne 0$, forcing $k=0$; the point is critical for $\pi$.

Thus the parameters with a singular equilibrium are exactly the critical values of this projection. Sard theorem makes that set a null set. Therefore almost every parameter gives a regular economy. No claim of existence at every parameter is needed for this sufficient regularity criterion.