  We present a new linearly implicit exponential integrator that preserves the
polynomial first integrals or Lyapunov functions for the conservative and
dissipative stiff equations, respectively. The method is tested by both
oscillated ordinary differential equations and partial differential equations,
e.g., an averaged system in wind-induced oscillation, the Fermi-Pasta-Ulam
systems, and the polynomial pendulum oscillators. The numerical simulations
confirm the conservative properties of the proposed method and demonstrate its
good behavior in superior running speed when compared with fully implicit
schemes for long-time simulations.
