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The $S1(1B1)\leftarrow S0(1A1)$ transition in pyrimidine has been investigated. High resolution laser induced fluorescence spectra were obtained by crossing a well collimated molecular beam with UV radiation of an intra-cavity frequency doubled ring laser. The extreme resolution enabled us to observe the molecular eigenstates of the excited electronic state. Most of the spectral features arising from the $S1$ state could be interpreted in terms of a slightly perturbed axis switched asymmetric rotor spectrum. Perturbations are found both in the energy levels of the $S1$ state and in the relative intensities of the transitions. This is in agreement with the much smaller background triplet level density in pyrimidime as compared to $pyrazine1$. Single level lifetime measurements have been carried out for rotational states up to $J\prime =4$. The results suggest a lengthening of the lifetimes of the excited state with increasing values of $K\prime$. The question put in the title will be discussed.