  We consider a system of differential equations and obtain its solutions with
exponential asymptotics and analyticity with respect to the spectral parameter.
Solutions of such type have importance in studying spectral properties of
differential operators. Here, we consider the system of first-order
differential equations on a half-line with summable coefficients, containing a
nonlinear dependence on the spectral parameter. We obtain fundamental systems
of solutions with analyticity in certain sectors, in which it is possible to
apply the method of successive approximations. We also construct
non-fundamental systems of solutions with analyticity in a large sector,
including two previously considered neighboring sectors. The obtained results
admit applications in studying inverse spectral problems for the higher-order
differential operators with distribution coefficients.
