Many of important equations in physics satisfy the least action principle, namely they are Euler-Lagrange equations for some Lagrangian. On the other hand it is known in mathematics that not all differential equations (both ODE and PDE) are Euler-Lagrange.
Are there physically important (!) equations which are not Euler-Lagrange?
"Physically important" depends on who you ask, but Type IIB supergravity theory in 10 dimensions does not have a simple action principle that is Lorentz-invariant. The standard reference for this is
Neil Marcus & John H. Schwarz, "Field theories that have no manifestly Lorentz-invariant formulation." Physics Letters B115, 111–114 (1982).
However, it is possible to construct an action for this theory by introducing auxiliary fields and/or breaking Lorentz invariance. See the following reference for more details and some historical references:
Ashoke Sen, "Covariant action for type IIB supergravity." Journal of High Energy Physics 2016:17 (2016).
