The authors deal with the radar ambiguity functions and their symmetries. It is well known that Hermite functions give rise to elliptic symmetries. Hermite functions are eigenvectors of the harmonic-oscillator Schrodinger operator. It is shown that the situation is essentially the same for hyperbolic symmetries: the signals are eigenvectors of the Schrodinger operator associated to the hyperbolic oscillator. Since this operator has continuous spectrum, these symmetries can only be reached approximately. It is also shown how to construct such signals and their ambiguity functions, which fall in two classes, are given. This work is essentially abstract nilpotent harmonic analysis, and is based on the well-known main fact that the Woodward ambiguity function is a positive-type function on the real Heisenberg Lie group.>