here Uj is an unknown second-order square matrix, and the Pj(x) are irreducible homogeneous polynomials of degree pj in x = (x1, x2, . . . , xm+1) ∈ CP. Suppose that system (1) has no Fuchsian singularities at infinity. Then the differential matrices Uj of system (1) must satisfy the condition ∑n j=1 pjUj = 0. By [1, 2], Eq. (1) is completely integrable if and only if the differential form (2) satisfies the condition ω ∧ ω = 0, (3) where ∧ is the operator of exterior product of differential forms. It was shown in [3] that conditions (3) are necessarily satisfied if the manifolds Mj, j = 1, . . . , n, form a pencil of algebraic surfaces. Then there exist two polynomials Q and R such that Pj = αjQ+ βjR for all j = 1, . . . , n, where αj , βj ∈ C, and Eq. (1) can be reduced to the onedimensional system