A series of Siamese twin designs intersecting in a BIBD and a PBD

Let p and 2p − 1 be prime powers and p ≡ 3 (mod 4). We describe a construction of a series of Siamese twin designs with Menon parameters (4p, 2p − p, p − p) intersecting in a derived design with parameters (2p − p, p − p, p − p − 1), and a pairwise balanced design PBD(2p − p, {p, p − p}, p − p − 1). When p and 2p − 1 are primes, the derived design and the pairwise balanced design are cyclic. Further, these two Menon designs with parameters (4p, 2p − p, p − p) lead to amicable regular Hadamard matrices of order 4p.