Solving $x e^{-(x-\lambda)^2}=c$ for various $\lambda$

Question

Suppose $\lambda$ and $c$ are real constants. I'm wondering if there's any special function that permits one to solve for the (real) variable $x$ in equations of the form $$x e^{-(x-\lambda)^2} = c$$ I'm familiar with the Lambert $W$ function and the solutions to the equation $x e^{-x^2} =c$, but I don't think that $W$ is involved in the general case I'm considering because of the presence $\lambda$, which alters the shape of the function $x e^{-(x-\lambda)^2}$ quite radically. Ideally, I'm looking for some sort of family of functions like $W$ that are parametrized by $\lambda$ -- does anyone know of references that might tackle this?