Embedded (4, 5) pairs of explicit 7-stage Runge–Kutta methods with FSAL property

The general case of embedded (4, 5) pairs explicit 7-stage Runge–Kutta methods with FSAL property ( $$a_{7 j} = b_{ j}$$ , $$1 \le j 7$$ $$c_7 1$$ ) is considered. Besides exceptional cases, the form five 4-dimensional families. within two (already known) families satisfy simplifying assumption $$\sum _j a_{ij}c_{j} c_i^2 / 2$$ $$i \ge 3$$ .