The next example comes from Hodge theory. On the differential forms in an n-dimensional oriented riemannian manifold M, we have two operations: the exterior derivative,
a first-order differential operator taking p-forms to (p+1)-forms, and the Hodge star
Composing these two operations we can form new operations; for example, the exterior divergence
and the Hodge lapacian
A form ω satisfying Δω=0 is said to be harmonic. Clearly the harmonic condition is second order.
Let us now define the Dirac-like operator D ≔ d + δ which does not respect the degree of the form, but takes forms of odd degree to forms of even degree and viceverse. Clearly if a form obeys Dω=0 it is harmonic; so again we have a first-order condition implying a second-order condition. (Of course, on a compact manifold the two conditions are equivalent.)
The underlying supersymmetric theory in this case is the one-dimensional supersymmetric sigma model. This is a theory of harmonic maps from the line (or the circle) to the manifold M. Upon quantisation, the hamiltonian becomes the Hodge laplacian Δ and the supercharges play the role of D.
When M has further structure, e.g., when M is Kähler, the sigma model has "more" supersymmetry and this supersymmetry underlies much of the Hodge-Lefschetz theory. Something analogous happens when M is hyperkähler.