Summary: Regularity properties of solutions to variational problems are established for a broad class of strictly convex splitting type energy densities of the principal form \(f : \mathbb{R}^2 \rightarrow \mathbb{R}\), \(f (\xi_1, \xi_2) = f_1( \xi_1) + f_2( \xi_2)\), with linear growth. We show that, regardless of the corresponding property of \(f_2\), the assumption \((t \in \mathbb{R})\) \[c_1(1+|t|)^{-\mu_1}\le f''_1 (t)\le c_2,\quad 1<\mu_1<2,\] is sufficient to obtain higher integrability of \(\partial_1u\) for any finite exponent. We also include a series of variants of our main theorem. In the case \(f : \mathbb{R}^n \rightarrow \mathbb{R} \), similar results hold with the obvious changes in notation.