I do not understand how the lagrangian
L
can correspond to two such oscillators.

You wrote two equations above, one is the equation of h.o. with friction, the other one has friction term with opposite sign, which leads to runaway.
The energy here is defined as
[tex]
E = p_x \dot x + p_y \dot y  L = m \dot x\dot y.
[/tex]
and since the Lagrangian does not depend on time, it should be constant in time.
You can verify this by multiplying the solutions for [itex]\dot x, \dot y[/itex]  the exponentials will cancel out and the result does not depend on time.
However, all this seems very artificial  I would like to see some useful application of it.