  A (1,{\lambda})-embedded graph is a graph that can be embedded on a surface
with Euler characteristic {\lambda} so that each edge is crossed by at most one
other edge. A graph G is called {\alpha}-linear if there exists an integral
constant {\beta} such that e(G') \leq {\alpha} v(G')+{\beta} for each
G'\subseteq G. In this paper, it is shown that every (1,{\lambda})-embedded
graph G is 4-linear for all possible {\lambda}, and is acyclicly
edge-(3{\Delta}(G)+70)-choosable for {\lambda}=1,2.
