  In this paper, two new families of fourth-order explicit exponential
Runge--Kutta (ERK) methods with four stages are studied for solving first-order
differential systems $y'(t)+My(t)=f(y(t))$. By comparing the Taylor series of
the exact solution, the order conditions of these ERK methods are derived,
which are exactly identical to the order conditions of explicit Runge--Kutta
methods, and these ERK methods reduce to classical Runge--Kutta methods once
$M\rightarrow \mathbf{0}$. Moreover, we analyze the stability properties and
the convergence of the new methods. Several numerical examples are implemented
to illustrate the accuracy and efficiency of these ERK methods by comparison
with standard exponential integrators.
