  We prove that the recently developed semiexplicit symplectic integrators for
non-separable Hamiltonian systems preserve any linear and quadratic invariants
possessed by the Hamiltonian systems. This is in addition to being symmetric
and symplectic as shown in our previous work; hence it shares the crucial
structure-preserving properties with some of the well-known symplectic
Runge--Kutta methods such as the Gauss--Legendre methods. The proof follows two
steps: First we show how the extended Hamiltonian system proposed by Pihajoki
inherits linear and quadratic invariants in the extended phase space from the
original Hamiltonian system. Then we show that this inheritance in turn implies
that our integrator preserves the original linear and quadratic invariants in
the original phase space. We also analyze preservation/non-preservation of
these invariants by Tao's extended Hamiltonian system and the extended phase
space integrators of Pihajoki and Tao. The paper concludes with numerical
demonstrations of our results using a simple test case and a system of point
vortices.
