Let $X \subseteq \mathbb{C}^n$ be a complex affine variety and $\tilde{X} \to X$ a surjective proper morphism where $\tilde{X}$ smooth. Is it true that every morphism $\mathbb{C} \to X$ can be lifted to $\mathbb{C} \to \tilde{X}$?

I'm particularly interested in the case where $X$ has an isolated singularity and $\tilde{X} \to X$ is a resolution.

Let $X \subseteq \mathbb{C}^n$ be a complex affine variety and $\tilde{X} \to X$ a surjective proper morphism where $\tilde{X}$ is smooth. Is it true that every morphism $\mathbb{C} \to X$ can be lifted to $\mathbb{C} \to \tilde{X}$?

I'm particularly interested in the case where $X$ has an isolated singularity and $\tilde{X} \to X$ is a resolution.