Invariant Ricci-flat K\"ahler metrics on tangent bundles of compact symmetric spaces

We give a description of all $G$-invariant Ricci-flat K"ahler metrics on the canonical complexification of any compact Riemannian symmetric space $G\mathrm{/}K$ of arbitrary rank, by using some special local $(1,0)$ vector fields on $T(G\mathrm{/}K)$. As the simplest application, we obtain the explicit description of the set of all complete $\mathrm{S}\mathrm{O}(3)$-invariant Ricci-flat K"ahler metrics on $T{\mathbb{S}}^2$, which includes the well-known Eguchi-Hanson-Stenzel metrics and a new one-parameter family of metrics.