  In this paper, we study rotationally symmetric extremal K\"ahler metrics on
$\mathbb C^n$ ($n\geq 2$) and $\mathbb C^2 \backslash\{0\}$. We present a
classification of such metrics based on the zeros of the polynomial appearing
in Calabi's Extremal Equation. As applications, we prove that there are no
$U(n)$ invariant complete extremal K\"ahler metrics on $\mathbb C^n$ with
positive bisectional curvature, and we give a smooth extension lemma for $U(n)$
invariant extremal K\"ahler metrics on $\mathbb{C}^n\backslash\{0\}$. We
retrieve known examples of smooth or singular extremal K\"ahler metrics on
Hirzebruch surfaces, bundles over $\mathbb{CP}^1$, and weighted complex
projective spaces. We also show that certain solutions on $\mathbb
C^2\backslash\{0\}$ correspond to new complete families of
constant-scalar-curvature K\"ahler and strictly extremal K\"ahler metrics on
complex line bundles over $\mathbb{CP}^{1}$ and on $\mathbb
C^2\backslash\{0\}$.
