Consider the Dirac equation $(i\gamma^{\mu}\partial_{\mu}-m)\psi = 0$. As it is well known, there are different representations for the matrices $\gamma^{\mu}$, $\mu = 0,1,2,3$, the most famous ones being the Dirac and the Weyl representations. It is not difficult to see that: \begin{eqnarray} \psi_{\pm} = e^{-iE({\bf{p}})t} P_{\pm}\phi({\bf{p}}) \tag{1}\label{1} \end{eqnarray} are (plane wave) solutions of the Dirac equation, where $E({\bf{p}}) = \sqrt{m^{2}+|{\bf{p}}|^{2}}$, $\phi({\bf{p}})$ are arbitrary vectors on $\mathbb{C}^{4}$, and $P_{\pm}$ is a projection operator to the subspace $\operatorname{Ker}(H_{D}({\bf{p}})\pm E({\bf{p}}))$ where $H_{D}$ is the associate Dirac (Hamiltonian) operator: $$H_{D}({\bf{p}}) := \sum_{j=1}^{3}\alpha_{j}p_{j} + \beta m \quad \alpha_{j} := \begin{pmatrix} 0 & \sigma_{j} \\ \sigma_{j} & 0 \end{pmatrix} \quad \beta = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix} $$ with $I$ being the $2\times 2$ identity matrix and $\sigma_{j}$ the $j$-th Pauli matrix. This is a general solution in which, if I'm not mistaken, is independent of the chosen representations for $\gamma^{\mu}$.
Next, we have another general solution for this equation, which is commonly used in QFT books in order to introduce the quantization of the Dirac field and which is given by: \begin{eqnarray} \psi(t,x) = \frac{1}{(2\pi)^{3}}\int_{\mathbb{R}^{3}}\frac{1}{\sqrt{2E({\bf{p}})}}\sum_{s=\pm}\bigg{(}a_{{\bf{p}},s}u_{s}({\bf{p}})e^{i(-Et+{\bf{p}}\cdot{\bf{x}})} + b^{\dagger}_{{\bf{p}},s}v_{s}({\bf{p}})e^{i(Et-{\bf{p}}\cdot{\bf{x}})}\bigg{)}d^{3}p \tag{2}\label{2} \end{eqnarray} where $a_{{\bf{p}},s}$, $b^{\dagger}_{{\bf{p}},s}$ are coefficients and $u_{\pm}({\bf{p}})$, $v_{\pm}({\bf{p}})$ are linear independent vectors which generate $\mathbb{C}^{4}$.
Let me remark that the in books I've consulted the general solution (\ref{2}) was constructed explicitly by using the Weyl representation. So, maybe (\ref{2}) is a general solution given the Weyl representation while the first one is representation independent? Many actual calculations done in textbooks use the Weyl representation, so (\ref{2}) would obviously be useful in this case.
My question is really straightforward: is there (and if so, what is) the connection between these two solutions (\ref{1}) and (\ref{2})? Maybe the vectors $u_{{\bf{p}},\pm}$ and $v_{{\bf{p}},\pm}$ must be eigenstates of $H_{D}({\bf{p}})$? Or are these solutions not connected at all?
