  We consider the system of two material points that interact by elastic forces
according to Hooke's law and their motion is restricted to certain curves lying
on the plane. The nonintegrability of this system and idea of the proof are
communicated. Moreover, the analysis of global dynamics by means of Poincar\'e
cross sections is given and local analysis in the neighborhood of an
equilibrium is performed by applying the Birkhoff normal form. Conditions of
linear stability are determined and some particular periodic solutions are
identified.
