  We define linearly reducible elliptic Feynman integrals, and we show that
they can be algorithmically solved up to arbitrary order of the dimensional
regulator in terms of a 1-dimensional integral over a polylogarithmic
integrand, which we call the inner polylogarithmic part (IPP). The solution is
obtained by direct integration of the Feynman parametric representation. When
the IPP depends on one elliptic curve (and no other algebraic functions), this
class of Feynman integrals can be algorithmically solved in terms of elliptic
multiple polylogarithms (eMPLs) by using integration by parts identities. We
then elaborate on the differential equations method. Specifically, we show that
the IPP can be mapped to a generalized integral topology satisfying a set of
differential equations in $\epsilon$-form. In the examples we consider the
canonical differential equations can be directly solved in terms of eMPLs up to
arbitrary order of the dimensional regulator. The remaining 1-dimensional
integral may be performed to express such integrals completely in terms of
eMPLs. We apply these methods to solve two- and three-points integrals in terms
of eMPLs. We analytically continue these integrals to the physical region by
using their 1-dimensional integral representation.
