  Paper is published in J. Phys. A: Math. Theor. 43 (2010) 225001,
doi:10.1088/1751-8113/43/22/225001. Exact analytical solution for the universal
probability distribution of the order parameter fluctuations as well as for the
universal statistical and thermodynamic functions of an ideal gas in the whole
critical region of Bose-Einstein condensation is obtained. A universal
constraint nonlinearity is found that is responsible for all nontrivial
critical phenomena of the BEC phase transition. Simple analytical
approximations, which describe the universal structure of the critical region
in terms of confluent hypergeometric or parabolic cylinder functions, as well
as asymptotics of the exact solution are derived. The results for the order
parameter, all higher-order moments of BEC fluctuations, and thermodynamic
quantities, including specific heat, perfectly match the known asymptotics
outside critical region as well as the phenomenological renormalization-group
ansatz with known critical exponents in the close vicinity of the critical
point. Thus, a full analytical solution to a long-standing problem of finding a
universal structure of the lambda-point for BEC in an ideal gas is found.
