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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 aclassification of such metrics based on the zeros of the polynomial appearingin Calabi's Extremal Equation. As applications, we prove that there are no$U(n)$ invariant complete extremal K\"ahler metrics on $\mathbb C^n$ withpositive bisectional curvature, and we give a smooth extension lemma for $U(n)$invariant extremal K\"ahler metrics on $\mathbb{C}^n\backslash\{0\}$. Weretrieve known examples of smooth or singular extremal K\"ahler metrics onHirzebruch surfaces, bundles over $\mathbb{CP}^1$, and weighted complexprojective spaces. We also show that certain solutions on $\mathbbC^2\backslash\{0\}$ correspond to new complete families ofconstant-scalar-curvature K\"ahler and strictly extremal K\"ahler metrics oncomplex line bundles over $\mathbb{CP}^{1}$ and on $\mathbbC^2\backslash\{0\}$.

Rotationally Symmetric Extremal K\"ahler Metrics on $\mathbb C^n$ and $\mathbb C^2\setminus \{0\}$