  We give first an easy construction of surfaces with $p_g=q=2, K^2=5$ and
Albanese map of degree $3$, describing an irreducible connected component of
the moduli space of surfaces of general type, which we show to be the only one
of the Main Stream with these invariants and satisfying a mild condition.
  We call it the family of CHPP surfaces, since it contains the family
constructed by Chen and Hacon,
  and coincides with the one considered by Penegini and Polizzi.
  We also give an easy construction of an irreducible connected component of
the moduli space of surfaces of general type with $p_g=q=2, K^2=6$ and Albanese
map of degree $4$, which we call the family of PP4 surfaces
  since it contains the family constructed by Penegini and Polizzi.
  Finally, we answer a question posed by Chen and Hacon, via three families of
surfaces with $p_g=q$
  whose Tschirnhaus module has a kernel realization with quotient a nontrivial
homogeneous bundle.
  Two families have $p_g=q=3$, the third is a new family of surfaces with
$p_g=q=2, K^2=6$ and Albanese map of degree $3$ (the existence of this family
is based on arXiv:2212.14877, joint work of the second author with Edoardo
Sernesi) .
