We construct examples of non-autonomous planar systems of the form $ dz/dt=z^{n}+p_{n-1}(e^{it})z^{n-1}+\ldots +p_{0}(e^{it}),\;p_{j}$ -- polynomials, which do not have periodic solutions. For the Riccati system $dz/dt=z^{2}+re^{it}$ we express the values $r_{j}$ of the parameter for which there is no periodic solution by means of zeroes of some Bessel function.