  Exponential integrators are explicit methods for solving ordinary
differential equations that treat linear behaviour exactly. The stiff-order
conditions for exponential integrators derived in a Banach space framework by
Hochbruck and Ostermann are solved symbolically by expressing the Runge--Kutta
weights as unknown linear combinations of phi functions. Of particular interest
are embedded exponential pairs that efficiently generate both a high- and
low-order estimate, allowing for dynamic adjustment of the time step. A key
requirement is that the pair be robust: if the nonlinear source function has
nonzero total time derivatives, the order of the low-order estimate should
never exceed its design value. Robust exponential Runge--Kutta (3,2) and (4,3)
embedded pairs that are well-suited to initial value problems with a dominant
linearity are constructed.
