  It is shown that a 2D system of free fermions on a square lattice with hoping
between more than nearest neighbours undergoes a fundamental electronic
topological transition (ETT) at some electron concentration n_c. The point
\delta=\delta_c, T=0 (\delta=1-n) is an exotic quantum critical point with
several aspects of criticality. The first trivial one is related to
singularities in thermodynamic properties. An untrivial and never considered
aspect is related to the effect of Kohn singularity (KS) in 2D system: this
point is the end of a critical line T=0, \delta>\delta_c of static KS in free
fermion polarizability. This ETT is a motor for anomalous behaviour in the
system of interacting electrons. The anomalies take place on one side of ETT,
\delta<\delta_c, and have a striking similarity with the anomalies in the high
T_c cuprates in the underdoped regime. The most important consequence of ETT is
the appearance of the line of characteristic temperatures, T^*(\delta) \propto
\delta_c-\delta, which grows from the point \delta=\delta_c on the side
\delta<\delta_c. Below this line the metal state is anomalous. Some anomalies
are considered in the present paper. The anomalies disappear in the case
t'=t''=... \to 0.
