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Explicit solutions for extended objects of a Q-ball type were foundanalytically in a model describing complex scalar field with piecewiseparabolic potential in (3+1)- and (1+1)-dimensional space-times. Such apotential provides a variety of solutions which were thoroughly examined. Itwas shown that, depending on the values of the parameters of the model andaccording to the known stability criteria, there exist stable and unstablesolutions. The classical stability of solutions in (1+1)-dimensional space-timewas examined in the linear approximation and it was shown explicitly that thespectrum of linear perturbations around some solutions contains exponentiallygrowing modes while it is not so for other solutions.

Analytic Q-ball solutions and their stability in a piecewise parabolic potential