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It is shown that every closed rotation and translation invariant subspace $V$ of $C\left({\mathbf{R}}^n\right)$ or $\delta \left({\mathbf{R}}^n\right)$, $n\ge 2$, is of spectral synthesis, i.e. $V$ is spanned by the polynomial-exponential functions it contains. It is a classical problem to find those measures $\mu$ of compact support on ${\mathbf{R}}^2$ with the following property: (P) The only function $f\in C\left({\mathbf{R}}^2\right)$ satisfying ${\int }_{{\mathbf{R}}^2}f\circ \sigma d\mu =0$ for all rigid motions $\sigma$ of ${\mathbf{R}}^2$ is the zero function. As an application of the above result a characterization of such measures is obtained in terms of their Fourier-Laplace transforms....