  We use generalized Gaussian quadratures for exponentials to develop a new ODE
solver. Nodes and weights of these quadratures are computed for a given
bandlimit $c$ and user selected accuracy $\epsilon$, so that they integrate
functions $e^{ibx}$, for all $|b|\le c$, with accuracy $\epsilon$. Nodes of
these quadratures do not concentrate excessively near the end points of an
interval as those of the standard, polynomial-based Gaussian quadratures. Due
to this property, the usual implicit Runge Kutta (IRK) collocation method may
be used with a large number of nodes, as long as the method chosen for solving
the nonlinear system of equations converges. We show that the resulting ODE
solver is symplectic and demonstrate (numerically) that it is A-stable. We use
this solver, dubbed Band-limited Collocation (BLC-IRK), in the problem of orbit
determination. Since BLC-IRK minimizes the number of nodes needed to obtain the
solution, in this problem we achieve speed close to that of explicit multistep
methods.
