= Solution
After division by $z$, the coefficient functions are
$$
P(z)=\frac1z-1,
\qquad Q(z)=\frac\lambda z.
$$
Both $zP(z)=1-z$ and $z^2Q(z)=\lambda z$ are analytic at zero, so the <regular singular point criterion for a second-order equation> shows that $z=0$ is regular singular. Substitution of $y=z^r$ into the leading terms gives the indicial equation
$$
r(r-1)+r=r^2=0.
$$
The exponent is repeated. Directly substituting $y=\sum_{n\geq0}a_nz^n$ gives
$$
(n+1)^2a_{n+1}+(\lambda-n)a_n=0,
\qquad
\boxed{a_{n+1}=\frac{n-\lambda}{(n+1)^2}a_n}.
$$
Thus $a_0$ determines the entire power series, so there is only one such solution $y_1$ up to scale. The <logarithmic solution from a repeated Frobenius exponent> predicts
$$
\boxed{y_2(z)=y_1(z)\log z+\text{a power series}},
$$
so its leading nonanalytic term is proportional to $\log z$.
For the contour ansatz, differentiation under the integral and one integration by parts give
$$
\begin{aligned}
0={}&\int_\gamma e^{zt}
\{z(t^2-t)+(t+\lambda)\}f(t)\,dt\\
={}&\left[e^{zt}t(t-1)f(t)\right]_{\partial\gamma}
+\int_\gamma e^{zt}
\left\{(t+\lambda)f-[t(t-1)f]'\right\}dt.
\end{aligned}
$$
The integral vanishes when
$$
t(t-1)f'+(t-1-\lambda)f=0,
$$
whose solution is the <Laguerre contour-integral amplitude>
$$
\boxed{f(t)=t^{-\lambda-1}(t-1)^\lambda}.
$$
The contour and branches must make $f$ single-valued along the traversed path and must kill the endpoint term
$$
\left[e^{zt}t^{-\lambda}(t-1)^{\lambda+1}\right]_{\partial\gamma}.
$$
Now suppose $\lambda<0$ is nonintegral. Near $t=0$, $f(t)=O(t^{-\lambda-1})$ is integrable and the endpoint factor is $O(t^{-\lambda})\to0$. We may therefore choose
$$
\gamma_1:quad 0\longrightarrow
\text{one counterclockwise loop around }1
\longrightarrow0,
$$
a finite loop based at the branch point $0$ and avoiding $1$ except by encirclement. A second choice is
$$
\gamma_2:quad 0\longrightarrow-\infty
$$
along one bank of the negative real axis, with a consistent branch. At $-\infty$, $e^{zt}$ kills the algebraic endpoint factor because $\operatorname{Re}z>0$.
The $\gamma_1$ integral is analytic in $z$ because its contour is finite, and it is a nonzero solution. By uniqueness of the analytic local solution, it is a constant multiple of $y_1$. This is the <Finite Laguerre contour solution> construction.
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