  An excitation dynamics of new quantum systems of N equidistant energy levels
in a monochromatic field has been investigated. To obtain exact analytical
solutions of dynamic equations an analytical method based on orthogonal
functions of a real argument has been proposed. Using the orthogonal Legendre
functions we have found an exact analytical expression for a population
probability amplitude of the level n. Various initial conditions for the
excitation of N-level quantum systems have been considered.
