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Previously known exactly solvable models of 2D semiclassical dilaton gravityadmit, in the general case, only non-extreme black holes. It is shown thatthere exist exceptional degenerate cases, that can be obtained by some limitingtransitions from the general exact solution, which include, in particular,extremal and ultraextremal black holes. We also analyze properties of extremeblack holes without demanding exact solvability and show that for suchsolutions quantum backreaction forbids the existence of ultraextreme blackholes. The conditions,under which divergencies of quantum stresses in a freefalling frame can disappear, are found. We derive the closed equation withrespect to the metric as a function of the dilaton field that enables one,choosing the form of the metric, to restore corresponding Lagrangian. It isdemonstrated that exactly solvable models, found earlier, can be extended toinclude an electric charge only in two cases: either the dilaton-gravitationcoupling is proportional to the potential term, or the latter vanishes. Thesecond case leads to the effective potential with a negative amplitude and weanalyze, how this fact affects the structure of spacetime. We also discuss therole of quantum backreaction in the relationship between extremal horizons andthe branch of solutions with a constant dilaton.Comment: 31 pages. In v.2 typo in Ref. [2] corrected, 4 references added.  Accepted in Class. Quant. Gra

Exactly solvable models in 2D semiclassical dilaton gravity and extremal black holes