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We study the ordered "excitonic" states which develop around the quantumcritical point (QCP) associated with the electronic topological transition(ETT) in a 2D electron system on a square lattice. We consider the case ofhopping beyond nearest neighbors when ETT has an unusual character. We showthat the amplitude of the order parameter (OP) and of the gap in the electronspectrum increase with increasing the distance from the QCP, \delta_c - \delta,where \delta = 1-n and "n" is an electron concentration. Such a behavior isdifferent from the ordinary case when OP and the gap decrease when going awayfrom the point which is a motor for instability. The gap opens at "hot spots"and extends untill the saddle points (SP) whatever is the doping concentration.The spectrum gets a characteristic flat shape as a result of hybrydizationeffect in the vicinity of two different SP's. The shape of the spectrum and theangle dependence of the gap have a striking similarity with the featuresobserved in the normal state of the underdoped high-T$_c$ cuprates. We discussalso details about the phase diagram and the behaviour of the density ofstates.Comment: 15 pages, 14 EPS figures included, EPJ style included, added  references, changed conten

Some consequences of electronic topological transition in 2D system on a square lattice: Excitonic ordered states