Are there fifth-kind and sixth-kind generating functions? Background.
In Goldstein's Classical Mechanics (2nd ed.), Section 9-1, pgs. 382-385, the generating functions (hereafter denoted $F$) for canonical transformations are introduced. From here on out, I'll refer to the original canonical coordinate variable and canonical momenta variable as $q$ and $p$, while the "new" canonical coordinate and momenta variables which I'm transforming to will be $Q$ and $P$. I believe this is standard notation.
There are four separate cases enumerated in the text - when the canonical transformation is best described by a generating function dependent on only on the mixed coordinates:

*

*$q$ and $Q$ ~ the transformation relations can be found by setting $F=F_1(q,Q,t)$

*$q$ and $P$ ~ the transformation relations can be found by setting $F=F_2(q,P,t)-QP$

*$p$ and $Q$ ~ the transformation relations can be found by setting $F=F_3(p,Q,t)+qp$

*$p$ and $P$ ~ the transformation relations can be found by setting $F=F_4(p,P,t)-QP+qp$
From what I've read online, the four types of variable-dependencies for the generating function enumerated above are known as the four-kinds of the generating function (i.e. it seems well-defined as standard nomenclature in the literature).
My question is, how come the cases where the canonical transformations are best described by a generating function of non-mixed coordinates covered? (i.e. $F_5(q,p,t)$ or $F_6(Q,P,t)$) Are there no canonical transformations best described by such generating functions? Am I overseeing something trivial?

Example.
Suppose there was a canonical transformation that was best described by a generating function dependent only on $q$, $p$, and $t$. Then for the generating function I could use
$$F=F_5(q,p,t)-pQ$$
I could plug that into the fundamental relation between the original and newly transformed-to quantities to then get the transformation relations.
$$\begin{align*}\require{\cancel}
p\dot{q}-H&=P\dot{Q}-K+\frac{dF}{dt}\\
&=P\dot{Q}-K+\frac{\partial F_5}{\partial t}+\frac{\partial F_5}{\partial q}\dot{q}
+\frac{\partial F_5}{\partial p}\dot{p} - \dot{p}Q - p\dot{Q}\\
&\\
&\rightarrow \left(H-K+\frac{\partial F_5}{\partial t}\right)+\left(\frac{\partial F_5}{\partial q}-p\right)\dot{q}+\left(P-p\right)\dot{Q}+\left(\frac{\partial F_5}{\partial p}-Q\right)\dot{p}=0
\end{align*}$$
I can now read off the transformation relations as:
$$Q=\frac{\partial F_5}{\partial p}\tag{1}$$
$$p=\frac{\partial F_5}{\partial q}\tag{2}$$
$$P=p\tag{3}$$
$$K=H+\frac{\partial F_5}{\partial t}\tag{4}$$
I can see that (1) and (3) define the canonical transformation, whereas (2) puts a restriction on what functions $F_5(q,p,t)$ will work. Nevertheless, this still defines a family of canonical transformations, doesn't it?
 A: Two reasons


*

*A generating function by definition generates canonical transformations. If you take terms like $F_{5}(p,q,t)$ then it doesn't relate coordinates involving two separate canonical transformations.

*The area preserving property of canonical transformations imply the existence of generating function. Then there is no way you can construct generating functions like 
$F_{5}(p,q,t)$. This is because if you take the generating function to be $F_{5}(p,q,t)$ then there is no way in which
$$\oint_{C}\dfrac{\partial F_{5}(p,q,t)}{\partial q} dq + \oint_{C}\dfrac{\partial F_{5}(p,q,t)}{\partial p} dp$$ where C is an arbitrary closed curve in the phase space can be tinkered to give the differences in the phase space volume
$$ \oint_{C}p dq - \oint_{C}P dQ$$
by adding and subtracting 
$$ \oint_{C} d(pq) = 0$$ or
$$ \oint_{C} d(PQ) = 0.$$ 
A: *

*Because of the extra condition (3) OP's type 5 canonical transformation (CT) is the same as a type 2 CT with an extra condition (3), and hence nothing new.


*The 4 types of CTs are locally sufficient (via the implicit function theorem) in a $2n$-dimensional phase space if $n=1$. For $n>1$ there are various hybrid types.
