There are a number of publications on relativistic objects dealing either with black holes or naked singularities in the center. Here we show that there exist static spherically symmetric solutions of Einstein equations with a strongly nonlinear scalar field with potential $V(varphi)simsinh(varphi^{2n})$, which allow the appearance of singularities of a new type (spherical singularities) outside the center of isolated configuration. The space-time is assumed to be asymptotically flat. Depending on the configuration parameters, we show that the distribution of the stable circular orbits of test bodies around the configuration is either similar to that in the case of the Schwarzschild solution (thus mimicking an ordinary black hole), or it contains additional rings of unstable orbits.
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