Solving linear systems via Pfaffians

An explicit solution is given for the system of linear equations EφEt=φ′, where φ, φ′ are alternating matrices of Pfaffian 1, which at most differ in their first row and column, and E is of the form . If v, v′, w∈Rr+1 with 〈v,w〉=1=〈v′,w〉 then a sequence of Cohn transforms with respect to (the fixed) w which takes (v,w) to (v′,w) is prescribed.