**Bose-Einstein
Condensation of the **D**-dimensional System of the Ideal Bose-gas with
Excitation Spectrum ** e

an Arbitrary Profile

Asymptotical methods are used to derive the
criterion of BEC possibility, and the criteria determining the order of phase
transition at BEC and indicating that BEC can be a phase transition (PTn) of an arbitrary order v Î
[2,¥) with finite or infinite jumps of the k-th
(k = n - 1) derivatives of
chemical potential m(T) and total energy E(T) with
regard to T and that all this is being determined by parameter c = D/s++¼+ (q_{i} - are the exponents of
calibrating functions of asymptotical behavior of
at the minimum
point). Parameter c takes on an infinite sequence of
values (c_{k }= k/(k - 1)), setting the Dc interval in which the BEC is possible, and for each PTn Dc_{n} subintervals of Dc and on each Dc_{n} there are subintervals where the jumps of m(T) and E(T) are either finite or infinite. The universal
role of parameter c in systematization of infinite
variety of extended systems "external field-system of bosons" by classes of
identical nature of phase transitions at BEC is also shown.